[lbbrudsk] deriving the quaternion multiplication table

the diagram below describes how to derive the off-diagonal entries of the quaternion multiplication table from ijk = -1 .  apply the multiplication operation indicated by the label on each arrow to both sides of the equation, then use ii = jj = kk = -1 to annihilate factors.  quaternion multiplication is associative (by definition), and multiplication by -1 is commutative.  an exhaustive proof of consistency therefore has 16 parts, one for each arrow direction, which William Rowan Hamilton presumably worked out all in his head while walking by the bridge -- pretty impressive.

every arrow is double-ended: applying the same operation twice always brings you back to where you started.  this is a consequence of i^2 = j^2 = k^2 = -1.  applying the same operation twice equates to multiplication by -1, and then we cancel out the minus sign.

LaTEX source, using ticzcd (commutative diagrams):

\usepackage{tikz-cd}

\newcommand{\mytimes}{*} % or \cdot

\begin{document}

% the direction of the arrow affects which side of the arrow the label goes
\begin{tikzcd}[row sep=9ex]
& & & ijk = -1 \ar[ldd,bend left=10,"i\mytimes",leftrightarrow]\\
ii=-1 & ij = k \ar[r,"i\mytimes",leftrightarrow] \ar[rru,bend left=20,"\mytimes k",leftrightarrow] & ik = -j & \\
ji = -k \ar[rr,bend left=25,"j\mytimes" ,leftrightarrow] & jj = -1 & jk = i \ar[u,"\mytimes k",leftrightarrow] & \\
ki = j \ar[r,"k\mytimes",leftrightarrow] \ar[u,"\mytimes i",leftrightarrow] & kj = -i \ar[from=uu,bend left=45,"\mytimes j" near end,leftrightarrow, crossing over] & kk=-1 &
\end{tikzcd}

previously, multiplication table of the icosahedral group.

[kcaualfy] multiple-board chess with shared edges

consider a large cube made of congruent small cubes.  consider just the small cubes on the outer shell of the large cube, those touching the outside.  consider playing chess in those outer cubes.

it is mostly 2D chess among 6 boards, the 6 faces of the large cube.  we restrict movement during a turn to a face: no traveling unimpeded over an edge like a true 2D manifold.  if a piece starts in an edge cube, it is in 2 faces simultaneously: it can choose which face to move in.  (for a computer UI with 6 2D boards, if a piece moves from an edge to an internal square of one board, it disappears from the other board.  likewise, if a piece moves to an edge, it appears on another board.  the overlapping edges need to be identified, maybe by color.)  corner cubes are simultaneously part of 3 faces.  because moves are restricted to a face, moves which hop from face to face without stopping on an edge are forbidden.  (without this restriction, we might imagine a bishop or knight hopping from face to face bypassing an edge.)

you can only move one piece per turn: pick which board is most urgent.

conventional wisdom for chess, central squares are important, edges are weak, likely no longer holds.

size 8 cube = 296 surface cubes, or 4.625 larger than a standard chessboard

size 6 = 152 = 2.375x

size 4 = 56, but perhaps a little cramped?

generally, n^3 - (n-2)^3.

initial position of pieces and details of piece movement (especially pawns) remain to be specified.  checkmates become more difficult because you cannot use the sides or corners of a board to restrict movement of the opposing king.

bishops can change color.

more generally, we can glue together many boards through overlapping edge squares.  (previously, not overlapping.)  many boards, not just two, can overlap at an edge.  similar to abstract polyhedra.  a piece in an overlap region occupies many boards simultaneously.  tesseract topology might be interesting: 24 boards, 3 boards per edge, 6 boards per corner.

four boards:

W X
Y Z

bottom edge of W and top edge of Y overlap, populated with white's pieces in initial position.  bottom edge of X and top edge of Z overlap, similarly more white pieces.  top edge of W and bottom edge of Y overlap, black's pieces in initial position.  top edge of X and bottom edge of Z overlap, more black pieces.

right edges of W and Y overlap with left edges of X and Z, respectively.  left edges of W and Y overlap with right edges of X and Z, respectively.

approximately flat torus topology.

white king starts at bottom right corner of W, equivalently bottom left corner of X, top right corner of Y and top left corner of Z.  there is only one white king.  similarly, black king at top right corner of W, also occupying all 4 boards simultaneously.  whatever pieces (perhaps queens) that start at the top left corner and bottom left corner of W are also shared between all 4 boards.

assume 8x8 boards.  pawns start on 2nd and 7th ranks of all 4 boards.  white pawn a2 is also on h2 of a horizontally adjacent board, similarly a7 and h7.  because of sharing, each player starts with 28 pawns.  white pawns in W and X, starting on the second rank, advance upward.  white pawns in Y and Z, starting on the seventh rank, advance downward.  black pawns the other way.  pawn promotion is the other end of the board.  promotion is particularly powerful because promotion zones are simultaneously part of 2 boards (or 4 if corner).

4 boards, each n-by-n, has (2n - 2)^2 distinct squares.  196 squares for n=8.  intriguing is n=5, 64 squares, which starts with the same number of non-pawn pieces as orthodox chess but twice as many pawns.  opposing pawns start separated by only one rank.  lengthening each board to 5x8 restores 4 rank initial separation.  (2n-1)*(2m-1) distinct squares: 135 squares for 5x8.

even more generalization: each board can be any shape composed of squares (polyominoes, polyplets).  an overlap region between a pair of boards is a bijection between subsets of squares (not necessarily edge squares).

[owxanyzo] a song of fire then ice

red giant, then white dwarf: winter is coming, but we have to survive fire first.

if your local star weighs more than about 10^58 amu, the fire will be crazy violent: supernova.  fortunately, our sun is less, about 10^57 amu.

(the lowest mass red dwarfs are about 10^56 amu.)

[soamhbbu] convex uniform polytopes

all the regular polytopes in all dimensions have been enumerated (things become boring in 5 dimensions and higher), but it seems a similar enumeration of all convex uniform polytopes has not yet been accomplished.  interesting new symmetries are known to become possible in 6, 7, and 8 dimensions.

are there interesting polytopes in 24 dimensions or thereabouts?  certainly the Voronoi cell (cells?  are all cells congruent?) of the Leech lattice, though I have no idea if it or its dual is uniform.

[dwjayiwr] first factor mod 210

assign letters to prime numbers as follows:

a 11
b 13
c 17
d 19
e 23
f 29
g 31
h 37
i 41
j 43
k 47
l 53
m 59
n 61
o 67
p 71
q 73
r 79
s 83
t 89
u 97
v 101
w 103
x 107
y 109
z 113
A 127
B 131
C 137
D 139
E 149
F 151
G 157
H 163
I 167
J 173
K 179
L 181
M 191
N 193
O 197
P 199
Q 211
R 223
S 227
T 229
U 233
V 239
W 241
X 251
Y 257
Z 263

below is a table of smallest prime factors of numbers, omitting numbers divisible by 2, 3, 5, or 7.  primes are indicated with a period.

previously, numbers in rows of 100.  this time, each row spans a range of 210.

previously, marking composites and primes with white and black dots, respectively.

\     mod 210
                                 11 1111 1111 1111 1111 1111 1112
div    111 1223 3444 5566 7778 8900 0012 2333 4455 6667 7889 9990
210   1137 9391 7137 3917 1393 9713 7931 7179 3917 3793 9171 3799
    0 .... .... .... .... .... .... ...a .... a... ..b. ..a. ...a
    1 .b.. .... b.a. .... ..c. b... .ac. .a.. ..d. .b.. .c.. ba..
    2 ...d ...a .... a.b. .c.. .a.. ceb. .d.b .... a.d. ...b ...c
    3 .... a... ea.. .b.c .d.e ..c. a... ..b. .da. b.ca ..d. ....
    4 fe.. ..ab .... dfc. .a.b ...e .b.g ...a .e.. cd.. ..b. .c..
    5 ...a .fbe .... ..a. d..a cg.. bd.. a..f .a.c ..e. ...c af..
    6 bgd. .... .... b... agbc de.f .h.. dba. e.cb .... .a.. .g.b
    7 .... ...d a.ch .a.f e... ...a d..h .... .... e.ag cb.a ...e
    8 id.. .b.f c..a .h.. c..i f.b. ..a. b.ec .g.a d.jc a... ....
    9 g.a. e.dc i..b f..d hba. ..a. .... .j.. d.be .af. .dg. ....
   10 a..f ba.. ...d .c.a bi.h abg. .k.. ce.. .b.h g... j..f ..a.
   11 .aeb c... ..b. ce.. .... .f.d .i.a ...g a.e. ..hb dkai .eba
   12 ..cj .... .bac g.fb ..ed ...j ha.d .a.. .c.. .... .h.. .a..
   13 ..bi ..ga .ck. a... ..lf .ad. .c.. ..kd b.j. a.b. .i.e h.f.
   14 cb.. a... badf i..g .e.. b..c a.j. .hc. ..ad fb.a ..lg b.jk
   15 ef.. .da. ..ge ..b. .a.l ic.. ..b. fcda h... .g.. ..kb ..c.
   16 ...a gc.. jdi. .bae k.da .... ..em a.b. ga.. b... ...l a..j
   17 ...c h.mb .e.. .d.. a.ib .d.. .bf. ..a. k.n. .h.d eab. l...
   18 dc.. f.bh a..j .ae. ..c. l..a b.ck .... ...g ..am hc.a fie.
   19 b... d... .fha b... gcb. .n.. c.a. eb.. ..ia ...e aj.h klmb
   20 ..a. .i.. d..g ...c ..a. ..ab mgdf .n.. j.d. .ac. fbi. e.l.
   21 a..d jae. ..n. .ica ..o. a.b. ...e bd.. fk.. ced. b..j .cag
   22 .ai. .... .m.b ..gj .bh. cl.. f..a k.o. adbc .... ..ac ..na
   23 .kfh bej. g.a. d.om b..c .b.. .a.. .a.. .bc. .d.. ..f. .ak.
   24 p.gb .nha ..b. a... d..k ealh .d.b ..g. p.f. ai.b ce.. ..bf
   25 m.de a... cao. ..kb c.q. d..l aegi d..c ..a. ...a n... .b.l
   26 j.b. ..ac e... h... .afe g.o. d... h.fa bpgi .cbj .... ....
   27 lb.a ..i. b.f. mca. ...a bqfe l... a..h .a.. db.. .... a...
   28 .jp. c.de ng.. c.bd a.mo kj.g .lbc ..ab d.h. ..e. qa.b .m..
   29 ..cg i.f. a..c .a.k n.g. heia .... ..b. ecr. b.a. ...a n.d.
   30 ..m. p..b .c.a .... e..b ..hd jba. .mik c..a ef.. a.b. jqoe
   31 c.an .kbg ...r .... .fad ..ac b.hd .fcn ...m .a.i ..h. .d..
   32 al.. eacj f..o b..a ..b. acd. .... ib.d ...b .psn .o.. g.ab
   33 fal. .c.. ..d. .f.. .kj. ..rb g..a .eh. a.qd ikg. .ba. c..a
   34 h.ec .bop .ja. .ed. ...g ..b. .a.l badf .he. o..p b.c. .ai.
   35 .chl .qka sd.b ag.. ibc. ja.f ..cg .... m.b. a.q. .c.. d...
   36 .o.. a... pa.. ed.f bc.. .bkr a... ..j. .ba. ..ma p.ne ...c
   37 dgjb ..af hqb. ..ic .ak. f.c. ...b l..a i.t. ..cb ..qd .gbr
   38 en.a dl.. .bpe f.ab s..a .i.m .... a..e .akr c.fg i... ab.d
   39 .mbf .j.. d... .qhe a... c... .jd. .la. bgdc .nb. .a.c s..h
   40 gbkd .... ae.. ra.. jhnc bf.a ko.. .d.. .scj .ba. e.ga b...
   41 rh.. .tl. .ica ..b. .d.. ..g. e.a. ...b .d.a go.. amdb ..e.
   42 ..a. .h.l c... dbs. c.af mha. r... e.bc ...k batc .... .pf.
   43 a..s .a.b .kjf gn.a d..b a.e. .bi. ..tl .o.. fc.. .nb. e.a.
   44 .ad. kmbq ...h .cpi .o.. d... b.ka c..s ai.. .eu. ..a. ...a
   45 b... c..d l.a. bh.g .tb. ...i daqc na.j lf.b .m.. ..eg .a.b
   46 .dc. .e.a .tgc a..p h... .ajb ..f. ..ui .c.. ag.. .bj. m..p
   47 .i.. abd. .aek ...d .n.h e.b. acou bq.. cjah r..a bdt. f...
   48 c..e ..a. oflb ...q .a.. ...c ne.v m.ca .lbf ..h. .g.. .jd.
   49 i..a b.c. e... .ra. bj.a ub.d h.vf ac.. .alg .... dh.k a.c.
   50 .e.b oc.. isbl n.m. aupd ...e .w.b ..a. fe.. ..kb ma.. cdb.
   51 .p.c ...e ab.g ka.b .i.j .vda fgr. .h.d .... sqa. ..ca .b.n
   52 ocb. .g.k .uda ..r. f.c. vewq .ian .j.. b..a ..b. acfi ...g
   53 .bap .d.. b... lod. eca. bwak c... ..dm ..f. ea.t j... bk.c
   54 a..i ha.s gd.m ..ba .vd. a.c. .xbq ..eb .... .hcf .i.b dsa.
   55 .agj ep.h .o.. ibc. .mf. wdnj .txa ..b. a..e b..d h.am .cga
   56 drn. ...b k.a. .l.. ...b cpf. .agy .a.q ..jc ..r. ..bc .a..
   57 ..e. doba ..in ael. ...c ga.. bkjx .... ..co aln. .f.. .ejd
   58 bqt. a.f. dac. b..h ..b. ...p a.d. gbyu ..ab ..la ckst ...b
   59 ..rd ..a. cg.. em.. cah. ...b .f.. .d.a s... .fdc .b.e .i.j
   60 ...a .bqc ..k. ..al .dga ..b. up.. afk. .ai. .czl b.d. ao..
   61 e..v .ih. fp.b dcnr ab.. ...h ...o c.ae ..b. .d.. gai. ...k
   62 fs.. b..g ah.q cawe b... .b.a .dec ..m. .b.. ..ar ojka q...
   63 v.cb .f.t .eba hu.. kl.. d..o ..ab dg.f jc.a m..b a... gfbt
   64 ..a. j..d .bu. w.eb m.a. s.af dc.p .i.h cx.. nag. .l.j .be.
   65 adbr .a.. .... p..a ...g am.c ...k e.c. b.h. dibe ..v. yna.
   66 sa.. ..cf b... .g.d ..lq bce. .nta pc.. a..x hb.. vda. b.ca
   67 .... qcem .wad fpbo r... .gh. .abe .a.b nm.i jef. ..lb cady
   68 .g.c r.ia w... ab.. zgsl .aqd ..h. ..b. .k.. a..u d.cf ig..
   69 jc.t ae.b qa.. .... ..cb nf.. abcd k... ..au ..xa .cbl .dh.
   70 nk.. i.a. ..e. ..f. .a.. e.dz bm.. ..ha .g.s t..x .e.. l.kc
   71 bj.a .y.o ..d. b.ac p.ba ljc. .es. abiv .a.b ..c. r.g. a.fb
   72 ..h. .d.y e.mf .jc. a..e o.gb .u.. rva. .... c..i .a.n .c..
   73 .eow .b.. ad.. .a.t .qd. c.ba j... b..g .ezc ..ah b.ma d.li
   74 ..wk .r.e h..a gd.. ob.c .d.. ..a. ..n. .fba ized ar.. ....
   75 d.a. b.gj ..c. ..u. b.ap kbas v.fm .... eb.. .a.. ct.d xhi.
   76 a..b dam. c.b. osha c.jn a... ...b ...c .t.p e.Ab ..og fxad
   77 wa.. ..uc dbg. ...b yh.. p.l. iqda j.ek a.df .c.m .sa. .b.a
   78 .hbd ekn. ..a. .ci. ..yv j..l .a.f cas. b.ne p.b. f..q .aml
   79 kb.. ch.a b..A a... .dr. ba.. m..c qej. fd.. ab.. igdu b.wx
   80 l.co a... za.c debv .k.. .n.. ahb. ...b .cag .dpa ...b .etq
   81 ..f. ..a. .c.h zbj. dae. ..py .c.h .nba c.B. bti. ..fw ...o
   82 c.da .jkb .ing eham ...a d..c .b.. a.c. uaf. .... A.be a..f
   83 .x.q .gbd ..v. .... ae.s .ckt b.l. uca. ...j q..f .aro ..cg
   84 bdA. .c.i a..e bag. t.bh ..za ..i. ybfe ...b d.ak wp.a c..b
   85 .l.c xnd. g.fa ...d ...r .Bfb ..a. .... dika .jho abc. ....
   86 .ca. vb.r .ejd m... ..a. .oai h.c. b.g. yB.. .a.. bcp. ..d.
   87 avk. .af. ...b q.ea .bm. a..d c.gl ..ri .zb. .w.. d... hmac
   88 .a.l b.sx ...u j..c b.od gbc. .f.a eh.j abg. wfce k.a. pd.a
   89 ..mb l..u n.ai ..c. qfC. t.d. .a.b gaod h.sk cy.b .zjr eab.
   90 i... .Aea fbd. a.ob n..i ca.g s..e l..r Cj.c ae.. ...c nbpu
   91 f.bg ad.. iax. .fdA ..gc q... a.k. .pd. b.a. .hba .ve. uj..
   92 Db.n seah bdcx .... .ad. b... .... .l.a .... .b.v c..y bfBm
   93 ...a z..g c.e. .da. c..a edof qibj a.pb wa.. k..c .e.b a.y.
   94 d..e ..lc .Bqk .b.f a..j rs.. te.. .gaw m..v bcj. .a.d g.A.
   95 p..i d..b a... sa.h ...b f..a gb.. cjl. pv.. ..a. .iba ...d
   96 .e.. c.bn d.ta c.qz ..hg .k.e b.ac .w.l reda ..f. a..k ....
   97 btad ...e ..Cc bg.x ..az iuam ...g wb.. qc.b .ad. ..nf .Bob
   98 am.j .ah. lci. k..a xdq. ae.b Cc.A .D.. cd.t .... .bd. .ra.
   99 cap. .byk mhso d.f. egi. ..bc ...a b.c. a.j. ed.. boa. .gDa
  100 .... ..c. ylab h... dbxf .c.k .aj. hae. ..b. .n.g .m.. .acA
  101 ..d. bcoa .rlf a.t. b.nx dav. ...s d..h Bbie a... q... c.r.
  102 gf.b aitd jab. yk.. .... C... a.nb fe.. ..a. z..a ..c. ..bj
  103 uce. .ma. kb.l .eyb .ac. h.gw ..c. .k.a .fe. dp.. zc.. Dbs.
  104 ..ba ..d. B.rj .nad .cea ..h. ckf. aA.g ba.. .mb. ud.. a..c
  105 .b.. f..p b..d e..c a... b.c. ..h. oia. .rEl ubcp .ahe b.d.
  106 z... ..g. af.. labs Cet. .rma ..b. n..b j.qf cia. d..a .ch.
  107 e..z jsE. p.ka vb.g ...d c... xoad mube ...a b... a.Dc Bd..
  108 h.a. .q.b ..g. A..e .nab ..a. .beF ...d fhci laq. ..b. t.Ck
  109 a.h. gab. vec. .lma ..w. aAs. by.. .... g..d .... c.k. i.a.
  110 baf. ndv. cq.r b.d. c.bu .... e.Da .bdc ao.b jl.c .ga. ..ea
  111 ..s. i..c hda. n..u wodE .tib .am. ea.B k.fg .cle sb.p da.f
  112 j.v. .b.a .... ac.w g... .ab. ...k boim .... a..d bCFB eh.n
  113 d... a.e. .a.b c.hl .bf. .... agzc .xf. ..a. .e.a ...d kp.h
  114 tjc. bga. ..fc .w.. ba.. .bf. D.o. i..a .b.. .... t.em ..vd
  115 .hqa .e.. dcb. .jan l..a ...r Acdb a.Ex cady i.sb .f.v aub.
  116 c..d .hf. gbe. ...b al.. ehnc j... kda. x.A. C.d. la.. jbir
  117 .kbe o.cq a.Fw .a.p idGt .c.a .e.. .cg. bdmr .fay .lda ..cp
  118 .b.C .c.j b.wa dm.. .f.e b.DE i.ah .f.. .uxa .bn. ay.. b...
  119 oeac t.AB f... rhb. dpa. g.ae .db. ...b ieg. .aD. ..cb .t.z
  120 acdF .a.e .j.. .b.a hAcB ai.. ..c. dqb. ..v. b.e. icl. oya.
  121 .a.k mf.b .g.. ...q scpb jeug cb.a ..mf an.h x... .Ga. .f.a
  122 .d.g ..bz .oa. ..nc e.g. k.cf ba.. .aj. .q.E dxc. ..Bl sa.e
  123 b.j. .wda .i.z a.cd m.b. .a.. h... vbe. ds.b a... gd.. lc.b
  124 .y.p aoBf ta.d u.j. ...F cm.b ar.. C..k ..ac .q.a Gb.c h.d.
  125 .... ybai un.. f.s. .azc ..bd .ji. be.a ..c. n.f. b..C gl..
  126 kwea .p.m ..cb .eai jb.a D..v gH.d a..o habj r.g. c..f adl.
  127 E... b... c... .... akeg .bdi .n.q A.ac .b.D .k.c .ax. .ou.
  128 ...b h.pc a.b. eaf. ...m El.a .C.b z..i nF.d .cab ...a ..bw
  129 .i.. .dkh .bja .cdb Gevf .g.p .ma. cHdq z..a .u.C a... .bf.
  130 egam ct.F .dhe cy.. vgaD n.ao ..rc ...e b.u. fabs ...h dgx.
  131 abc. .a.. b.mc jdra ..ki ad.l .pe. fB.j .cDq .btd .... bwal
  132 da.. .... ic.. ..bh .... ..j. .cba ..tb afkn .rI. e.ab w..a
  133 c... d.q. .sav .be. .ihy ...c eaf. .ab. oj.. b..G ..gn .aed
  134 x.kh f.ca d..p aH.o .t.b .agn kbdm ec.. ..d. a..e .Abi fjc.
  135 .rzd acbv .a.q ...G uj.. ..eh aE.p .dng ..af ..da kw.. c...
  136 b..c .va. .h.. b... .ab. .... y.le .b.a .d.b .e.m fic. .E.b
  137 .cxa ..gs .k.. dAa. F.ca ...b o.cI a.C. fa.. .djw .be. asm.
  138 qlrx kb.o .... .pzg ac.. i.bA cdk. bjah .... F.w. ba.g .H.c
  139 ..d. .Bn. a.eb .a.c fb.q d.ca .s.. dy.D ..b. Ega. jefa ....
  140 ..oe b..d .m.a .tcr b.i. hb.H deal ...y gbfa c... a..A vc.f
  141 .dab ..x. eEbk ...m o.ae cnaj .z.b B.Fp ...c daub .gsc .ib.
  142 ae.. lad. qb.. ...a pIfc a..e .Jhr .nf. decg .G.t Bdh. .ba.
  143 mab. Fi.e x.cd o... g... .kf. ..ja l.u. aw.. y.b. c.ak .Ida
  144 .b.r .l.. bxag .iG. c... be.d .aAu .ahc e..z .bkc df.. ban.
  145 hsi. .gfa j... a.b. eB.d .aC. ..bd .lqb .hpA acoz y..b .d.e
  146 ..h. an.k .a.. .bg. rqmp u.d. af.. cibd ..a. bf.a ..y. .m..
  147 ..to c.ab g.dj cG.. .a.b sJ.k .b.c Dfva ..od .i.h n.bt .k..
  148 ..ca Cdbl f..c H.a. ...a p... b..i aed. .a.. G... .j.. a.go
  149 bze. BJ.. .c.. bex. arbC .... .cgv tbal cEeb psHq .a.. de.b
  150 c... jfi. a... Dah. B.e. gd.a .qvw .kcf ..g. ..ad rb.a if.h
  151 d... .bc. l.za e..j nhs. .cbf .kaD bc.. l.Fa .Ap. a.Id n.cm
  152 Chay dcjt .g.b .z.f .ba. .vag ..wK q... ..b. .a.o .kuH c..d
  153 a..c ba.f dl.e ...a b.g. abI. .wd. ..ie mbds j... .rc. ..aq
  154 .a.b .... ..bD fK.e ..c. .H.. phca .dk. al.. ..db gca. ..ba
  155 j..f ...g .bah ...b .co. A.w. cath ia.u .dl. ...j epdf Ca.c
  156 L.bq ...a ...l ahec .... xacm e.pB .goI b... adb. ..ks g.e.
  157 .b.. a..n ba.C ..c. dJ.. bj.. ad.. er.z ..aG cbga ..p. bci.
  158 ..dt ..a. mD.E Ijb. iarf cxes ..b. d..a k..c .... .qnb e.fJ
  159 .A.a ..ed .vof lba. .y.a ..x. dD.e a.b. .ac. beh. .m.. a...
  160 .d.. ...b ..c. quiB aF.b mgo. hb.. fta. i... d... cab. k.q.
  161 .gE. .ebj a.u. .a.d cg.. yi.a bx.. ..sc df.. lnac id.a hgp.
  162 b..v .mrc ..ea blJt qwbo eyE. ..a. .b.. Ak.a .cKg aer. ..db
  163 .uae f... .j.F .cl. ..a. ..ab .en. c... hg.C Hai. dbA. fJ..
  164 akB. ca.. ef.. c..a ...d a.b. K.yc bLpF ...f ..l. btg. mda.
  165 .ac. h..r .z.b .nwE .b.k ..de ...a s.jd acb. gh.u fna. ...a
  166 p.j. b.De .cao ..kl bL.s .b.. qai. marg cbGd ..el hov. .a..
  167 c..b .d.a ..b. a.di .z.. .a.c fAFb n.cC ei.. aIBb v..h K.b.
  168 ..fk ajc. .a.. t.mb e.d. zc.i aj.. .cx. .pa. e..a m.fr dbce
  169 .Bb. .ca. .... .dqg ja.. kd.. .u.E .Kea bGfj vLbd l..g cAtf
  170 dbpa eD.. bwg. ..ak .sha b... n.mz a... qa.e .b.f .lcd a..E
  171 .c.h d.sA w.Gi ..b. a.c. .... .LbC .eab gx.. .jy. Fa.b r..d
  172 i.e. pkh. a.fm nau. .cli ..fa c.d. o.bv ..d. bBa. .g.a .e.c
  173 k..d H.wb ih.a ..Fc tmeb rqc. sba. .d.. ..Ma ..cJ afbm ...n
  174 ..aD ..b. rG.. e.c. gdal ..a. bo.n h..j .d.. ca.. q.de yc..
  175 a.u. sa.. ...g b.Ba .eb. a.jC .fM. .b.h r..b .d.. ..jc ..ab
  176 ea.w .gkp .H.e ...n df.c ...b vdBa .f.e ajc. .Cxp .bau lq.a
  177 ..di .b.. fAa. .mge I.N. dsb. .aet ba.. .o.H h..x bi.. .a..
  178 fDn. Ez.a ce.b af.. cbk. tah. d... ...c G.b. a..c e... .l..
  179 .dg. af.c Ma.n ..e. b.Dv ib.. a.hj .ogf uba. dcna Kxh. .fe.
  180 w..b mya. Grb. .c.d .a.j ..Ff .Igb csma dC.. ..jb Hd.. ..bM
  181 .Nka cq.y .b.d c.ab z.ia gle. k.oc ajh. .ag. m.q. .L.. abd.
  182 h.b. .Ief o.sc .v.. aE.. f..d ...e g.at bc.. Beb. daNp xiws
  183 .bh. .... acry fanC .Dud bmla tc.d .... cJik .ba. ..ea bd..
  184 c..f oe.. .kva ..b. ..g. ..dc ..as .Ccb .r.a .FOh a.ib .p.l
  185 ..a. k.cm h.du .bI. ..a. eca. H.k. .cbA .mjd ba.. gew. ..cD
  186 atie .a.b .n.. ..da y..b a... lbj. E.d. O.z. n... ..b. chax
  187 JaHc v.b. ed.. .ohD ..de .... bl.a .gG. a.rt kz.. w.a. dnfa
  188 bcqA .... j.af bd.p .hb. .d.e gacP .aJ. .e.b figd .c.. ua.b
  189 df.m .F.a .o.r a.As .c.g .a.b c.li f... n... a.e. .b.d ..zc
  190 .xIK ab.q .amj .gtc .p.. .eby a..g b... efai .wca bE.k .v.d
  191 .l.. .oaC d..b ..c. ea.. ngr. Bhd. ..Ia .Abo c.k. .jmM icJe
  192 ngla b.G. ..Lh kEa. bgpa cbs. ...h ade. ja.c ..d. .v.c ag..
  193 .p.b ex.k zfb. .h.. ad.c F.iK ..ub yqao tdce ...b .adj NDb.
  194 B.sl ..m. abc. da.b h... .u.a ...f .ei. .gw. .daH cF.a .b..
  195 g.bp l.jx cK.a Be.. c.th ...n .dao ..Lc bqea ..bc a.g. .e.r
  196 .ba. ...c b..t .k.. ..a. b.a. f.E. dGn. wvyr gah. o.J. b.mi
  197 a.f. .a.d k..s ecba f.L. a.zo d.b. ck.b ...B iq.. .hfb tOa.
  198 ja.. c... ..x. cb.. .e.n .p.q .kJa Klb. a.f. bysj ..a. h.ia
  199 e.cu ..db Fmac ...d i.Eb ..H. .a.. Ia.e dc.. ...f .db. .aP.
  200 uj.. ..ba Ac.d aC.e ..f. .a.p bceq w.f. czn. aE.L ...g ..d.
  201 b... aG.l saf. bjio ..b. ..fc av.. .bc. i.ab .g.a d..y ...b
  202 mF.. g.a. .... .xe. .a.d .cvb ep.d Gc.a g... uh.M ibC. jdck
  203 t..a .bfh .pD. ..a. ...a ..bF .r.. an.d .aK. ..Ae bgk. az..
  204 .q.c ..Hj lCdb m..x abI. ..e. of.. .ua. l.bd .fi. .ach e.N.
  205 .c.. bdeo ai.p .ad. bfc. Kb.a w.ce .fd. kb.. rea. Dc.a smnL
  206 ..yb z.q. fdba ...h .cdt oMBw cgab j... .s.a J.Cb a.e. d.bc
  207 f.a. Ge.i Dbl. pdvb ..au jda. ..i. .m.. .lL. .acd ..E. kb.g
  208 a.bh tay. ..eq vMca o..x a... .... pNjf bilm cHb. ne.d qca.
  209 .aje d.hI bO.l .p.z .... b..f .era .... ak.c .b.D A.ac b..a
  210 ..gG .... dhaB o.bf .H.c .N.. .ab. kagb F.c. .... ..ob .agm
  211 qeAd ujva n.c. ab.P ...w fate .jgG hdb. .eJl ard. c... M.k.
  212 Q... a..b ca.i f.y. cdwb g... ab.. ...c mdaj ..ea ..b. nuO.
  213 i.vf q.ac t... d.k. pa.i .eAx b... gz.a e.h. .cms .u.f I.K.
  214 br.a ..N. ig.. bca. d.ba hfqg xd.. ab.n .aob ejrO .... a..b
  215 H.dg cP.. q.j. clfw aig. d.hb I..c d.a. .ut. z..n .a.. .Cvo
  216 .mc. eb.d as.c .al. L..f kBba dih. b..J .cpe .la. b.ha ..f.
  217 Pdr. .A.g mc.a jw.k .bMp ..y. .ca. .e.j cBba d.lE a..o ..h.
  218 cfai bHdn .... .e.d b.a. .bac .y.O fgck dbep .a.. .dj. ge..
  219 aGKb Dac. .Mbd i..a .qe. ac.. g..b xcNH .f.. .vgb C.n. ..a.
  220 kahz .c.s .bBw eI.b ...g io.d ..fa .x.E a..F pPtr d.ae cb.a
  221 .nbc fm.. ..a. ug.. le.d ...N La.d Jat. b.v. .kbh ..c. faA.
  222 eb.E ...a bf.e a... .lc. bad. .scj ...d v..f abpq lc.. b...
  223 .gDr a.k. .ad. J.be .cnj .Ip. aqbf .Fob xyad ..ja fl.b mgBc
  224 ..Q. .daw KeO. .bdc .a.. ..c. ..n. vjba f.iy b.cg e.sq E.uh
  225 .IFa .i.b .d.. ..a. rhda .z.. eb.A a... sax. c..k j.b. ace.
  226 ghfO rwb. ..o. .d.. a.C. cdP. b.z. e.a. L.kc .u.d .afc ...q
  227 b.ij wh.. a... bam. ..bc Hhea ..p. .b.. C.cb g.a. my.a eF.b
  228 .skQ d.e. .Jca ..M. .rP. ...b khae ni.g ..ja xe.f abpl .B.d
  229 .waq .b.. c.Ah gtK. c.ao ..a. .Gdh b.fc D.d. qa.c b.e. lyj.
  230 a..d Qagc ..fb .hCa .bv. a.fu ...i rd.m N.bk .cd. .... p.aK
  231 Da.. b... jkem .c.g bd.. eb.J n..a cA.. ab.i ...t Leag zloa
  232 s..b crf. ..a. c... um.h .... Gakb .a.. B..h .d.b xr.m iabz
  233 Iyc. g.Ja eb.c a..b d..e .a.. .d.L .pD. gc.M afh. .o.. .bq.
  234 Geb. aQ.. .aCv .... .fs. dlie ac.. df.. bea. k.ba Egx. .w.n
  235 cb.. .uad b..k A... qa.. bO.c d..n ..ca j.mg obe. ...x b...
  236 fdta jKcv .N.z .fa. g..a Bcl. ..b. ac.b ea.s d.Ri ...b a.cG
  237 po.. .cd. ..yg .b.d a.r. qk.l .gs. iDaf d... b... Pa.k cfxe
  238 F..c .gjb a..d .aH. ..zb ..na .b.. t.e. ...L i.a. .wba B.dg
  239 lcn. eEb. ..Ma kygf ..c. C..d bKa. o.m. .p.a jh.. ac.s ..iv
  240 b.a. A.Qf g.qn b..y icad f.a. cl.d .bu. ...b manw h... .d.b
  241 aRe. Oart ..hK fe.a m.JH a.cb io.u z.gd .Ne. ..cj .brh veat
  242 .a.f .b.Q .Ld. .scF ..ey .mb. A.ga b..B a..d co.. b.af DcHa
  243 .j.. pd.. R.ab ekdh Cb.r cf.. .aP. .ad. qnbc .... is.c La..
  244 .lx. b.Ia kd.. ajf. bedc .a.. ..t. gksM .bcw a.v. .... d...
  245 e.lb a... .abe .d.. .oSf .d.g ak.b ..re ..a. .pia c..z j.b.
  246 dH.g ..a. cbEf ...b cagm .qMh ..el .o.a .wO. f..c .kDd .b..
  247 .fba d..c .e.N C.aI .A.a RG.. .mS. aEB. ba.. ncbp e..r a..d
  248 .bzm l.xg b.k. hce. a.j. b... e.d. cyar .fd. tb.. .a.I bne.
  249 ..Nd c.z. ajmP cabi ...s ...a Fnbc edwb .iTK ..ae pCua gq.k
  250 ..c. fl.B x..a .b.. .d.. jEei g.av ..b. ncha b.g. aDd. e.F.
  251 ..a. o.eb .cp. d.zt k.ab h.a. .bve .lji c..f ha.. .Sb. .MG.
  252 aij. Iab. .Q.. .g.a dO.. a.hc bdLf ..cu k.q. y... f.eJ ..a.
  253 bad. .ec. r..i b.j. .sbA dc.. D.ha db.. a..b CR.F .tap ..ca
  254 ig.T .csd ..aO xo.. .g.i e..b da.N Aal. rtEn ..qm yb.P cah.
  255 .dfc .bBa i... a... f... .ab. .ewM b.hl ..sj a..g b.cn Rpm.
  256 hc.. a.d. eaJb ..xd .bce ...n awc. .... dga. ..Pa .cq. Hr.f
  257 geh. b.a. l..d tup. ba.M .bDe ci.. k.na lb.z .j.f J.gi ..dc
  258 .k.a soFe .mbQ Nqac .Sfa ..cd .U.b a.f. .a.o g.cb d... a.bC
  259 y..i .... hbf. ..cb ax.d Gef. ...d ..ag e... ct.. Oa.. .brp
  260 .ub. N... avl. gak. e..E csda S... .TAd bl.c e.a. .f.a Gh.e
  261 mbqy ..fJ b.da s.h. .pBc b.jt ..aH C.e. K.ca .b.. aAj. bo.h
  262 .zak ed.. ..cl ..bg tha. ..aP .fbo .Bdb .j.e DaTu c..b ....
  263 ah.v .a.q cdgI mb.a cfd. an.. .... Oebc .r.u bg.c o.FG dja.
  264 .ae. gh.b fyz. Qd.k .jmb .d.o .bqa LnE. a.el .c.d ..a. .e.a
  265 d... K.b. U.a. lc.. .we. DxOA ba.j ca.k ..i. .... .g.d .a.s
  266 b.mp cf.a ...h aP.. ..bj .axM .uRc .b.f ..rb aKjC ..ie .fnb
  267 k.c. a... da.c .h.q ge.U t..b a.ds .j.. ocam l..a .bvA ...I
  268 eLid .ba. PcFe .lwf ha.G ..b. zc.. bd.a cq.. .kd. bB.E ....
  269 c..a .g.f ...b .Nae Hb.a fp.c ..e. aic. .abh Ll.. n.d. a.sg
  270 ...j bBc. .eK. dQg. a... ybrj .... .caz .bD. vdf. eaH. ..c.
  271 ...b .cu. a.b. .aeS d... .yka edAb ...t m.jE ..ab .h.a c.b.
  272 V.dc ...o nb.a ..Qb ..k. df.. tFa. d.g. J..a ..me a.cR hbg.
  273 .ca. .sid .w.L ..f. nCa. o.ar dpcq .h.v bTk. .ab. Bczu e..H
  274 abo. .ae. bpt. .V.a lcGf a..m c..e .vC. h.g. db.. .O.. b.ac
  275 .akn i.d. ...f ..bc ol.F ..c. k.ba g.xb a..r f.c. ldab ..Ia
  276 Ef.. hew. mgad .bc. .BA. ...g .a.W fab. u..t bh.n kl.. .ad.
  277 .qsg ...a ..e. a... D.gb ca.d vbIp uJP. .f.c a.Si debc ....
  278 r..e a.b. .ah. p..Q ..lc m..U aefd i.HD jxa. .A.a gG.h .d.i
  279 b..w f.ag e.cM bRt. .abe ..d. r.k. pb.a .F.b i.o. c.lj f..b
  280 Aewa Bmt. cfds Tpah czua ...b .... ag.c .aId ..yc .bn. a.i.
  281 ...o .bjc C..q ..d. a.h. z.b. g..f b.a. E.o. kce. ba.l q...
  282 .n.h ..KN ad.b .a.v Qbdg AeCa ir.. c... e.b. j.a. .M.a d..o
  283 w... b.hu ...a cdi. bG.. lbmh f.ac ...p ib.a enxd a... y..e
  284 d.ab .... shbc ..S. fQa. .ga. .E.b m.e. Mc.. ra.b iHfd .lbu
  285 ag.B da.U .bvt hDLa .g.q aF.I .cn. h.RV c.fe ..kN .J.. ubad
  286 cab. q... d... krm. G..C .j.c ..da Dech a.d. .Tbf mxa. t..a
  287 .bed ..ck bia. Ke.. s.f. bcq. Ja.B .afN .ge. Rbd. .n.w bacO
  288 gW.. vc.a q.f. a.bM Fdep hafk j.b. ...b .d.. a..B ..db ck..
  289 Cv.c ay.i .a.. dbr. ..o. .tg. aHi. n.bm Asap bd.a .fce x...
  290 .c.. ..ab .E.m .k.i dacb pL.l .bcD ..oa .i.. SrJG wcb. Px.l
  291 e.da .zb. k..e g.aB Ncja du.i bf.. akqe .a.O pfU. I..m avhc
  292 b..s ..gd .j.y b..c afb. ..cV dke. jbai ...b ..c. ta.. CS.b
  293 hd.. n... aeoD .acg T... j..a .l.. .O.s .h.. cwa. eb.a .c.x
  294 f.h. FbdR H.ga nfed z..Q c.b. eAa. b.j. dPma wg.v ad.c .We.
  295 i.a. gf.. ..kb .m.. ybac ..a. .Tl. e.kf g.bJ Ka.e .... .fdn
  296 a.ru ba.. h.c. ..ja b.yo abed pq.n PI.. .b.v ..Gs cg.. eAak
  297 ua.b tje. cDb. .HEf ciOd ..K. .j.a ...c avwg .e.b .paq .dba
  298 ..l. m.Cc .ba. ..hb g..R fUd. .ap. qamd .E.j .cB. u.ei .aoh
  299 ..bL xe.a .sdg ac.V .h.. Sany .g.l c... b.zd aGbr ..p. ...q
  300 Xbnf ad.. baeo c.d. m..P b.t. aR.c ..dw U.aC sbLa .eKf b..g
  301 .Mce lha. .djc iFb. .adI .f.. .eb. .u.a .c.. A.nW ...b dH..
  302 .CTa ..ox ec.. .ba. J..a id.D .c.. awb. caF. b..d .... a.z.
  303 ceg. .l.b ..ih j... a..b ..vc .b.h w.aj .e.S .B.. .abd .sf.
  304 .o.. d.be aANf La.. r.iu .cja b.gI kc.C ynt. f.a. .qja ..cd
  305 bf.. rcD. d.xa b.nu h.bm ge.. .sa. fb.. ejda G.E. a..S cikb
  306 K.ac ..lV z..x q.B. e.ah TDab Mm.. gdp. .fih ead. .bc. .jqe
  307 ac.m .a.l Xg.E zJka Udc. a.bg ..c. b.e. .d.. .yhA bcd. ..a.
  308 pa.g et.H .nmb div. qbg. VQ.. co.a T..l aWbe nd.. rha. f..a
  309 ..ik bD.. .fa. vx.c bI.j Lbcw .a.. ra.. .bNf .oc. g.mF ha..
  310 ..db ...a l.b. aec. .... daz. On.b dht. lpe. a..b fu.y .cb.
  311 Ws.. arRd va.. HB.b .Fe. c.E. a..m .g.. foac Ui.a jr.c gby.
  312 .db. ..a. .l.J e... xa.c ..QC f..i ...a buc. d.bK ...e ....
  313 .bfa hkd. btc. G.ad fe.a b..j .... ao.P da.i Bb.m cdf. a..U
  314 e.v. p.ih c..d .gbE aXxw ...Q .Ybg ..ab ..fG ...c hatb i.df
  315 s.yA ...c a.hl Vaqe .kwx .gua mJe. MrbF .I.n bcaf dz.a ..j.
  316 .g.. i.Ob oe.a .cAL .gfb ..i. .bad c.f. q.Sa .p.. a.bn .dv.
  317 .Da. c.b. jmf. c.eh w.a. MHan bB.c ..id .C.l .a.g ..Wv .Kej
  318 a.c. oa.p y.dc bt.a ..b. a.k. Q.FE ebn. .c.b ...e .fIN ..ab
  319 ga.h zdf. uc.j ..d. .Pk. .qeb Tc.a i.d. a..s ..Vk .ba. e..a
  320 c... .be. pda. yw.C ..dn .Jbc .ase bac. ..kN ge.t bjr. da..
  321 ..M. ..ca .h.b ady. .b.. .a.L xX.. .c.g j.b. a..d .Ye. o.c.
  322 d.kV acn. fapG g.lz bD.r .bW. at.. h... .ba. .l.a F..d cq..
  323 fK.b d.ar .obw .f.j .amz e... i..b ...a v.G. .ulb kecX .mbd
  324 .c.a .fj. dbw. E.ab ..ca Nn.s .ec. a.rf iadk W... .c.g ab.D
  325 B.bd Uo.. ekgH Is.l ac.e hi.f cOC. vda. b.qo hgbl ia.t ...c
  326 Rb.. g... a.n. Ba.c .d.. bKca .Mk. xz.L gd.m Hba. .sda bp..
  327 jJ.. Eq.e A..a d.b. l..O f..u .ya. txsb ...a cdej aghb .cn.
  328 ..a. ...C .iG. fbp. dla. ceav Ed.. F.b. e..c bafP l..c .ohm
  329 ajdf .a.b .qYk ...a e.Zb aj.. .b.o d.hx ..c. e... zlbf rNae
  330 ha.. wLbd ..cg RjDp .J.. .f.. bgia ..e. ahP. ..A. c.a. ...a
  331 bdhX eg.O cGa. bqfi cEb. rkFo ja.K saYc .i.b d.mc S..k ja.b
  332 zwC. .uda r... a.gd vpRf .a.b ...Q .e.. dyx. ackh .bl. FDf.
  333 ..ez ab.j gaId kc.. ...l .Vbm a.t. b..i r.ay fM.a b... .ed.
  334 .fg. c.ak ...b c... nae. ...d P.Hc f.ga o.b. TI.. d..l nhgw
  335 .pca b... mj.c eLao b..a .b.k ..gd a... Ga.. xFuy .X.e ad.h
  336 i..b H..q Scb. W... ae.i g.d. .cfb .Rad c.g. Ox.b AaZD .Jb.
  337 ch.p f.sv abde .aNb .... m.Ba ..q. g.ce ...d t.an ...a fb..
  338 ..b. .dc. kf.a X.de Si.K .c.g o.au Qcd. b.sa ..b. a... wycY
  339 .bag .cTo bd.. M.j. ..aZ b.a. shzf .q.. .... .a.. eECi b...
  340 a..c .a.n .P.h .dba .... adA. ejbh .U.b f..H .m.d gkcb ..av
  341 daoi s.pg .CrB .b.T juc. .... f.ca e.bG aq.j b.Ke .cad m..a
  342 .yf. d..b L.a. i... fc.b ..ep ca.. .akS .r.I ..N. .tbx ea.c
  343 .n.. y.ba du.. a..c ...h iacl b.de m... .tdh aecw H.Y. ...f
  344 b..d aA.. .ai. bOc. I.bg F..q ap.* .bG. .Mab cnda D.e. zcxb 121:269
-----------------------------------------------------------------
                                 11 1111 1111 1111 1111 1111 1112
mod    111 1223 3444 5566 7778 8900 0012 2333 4455 6667 7889 9990
210   1137 9391 7137 3917 1393 9713 7931 7179 3917 3793 9171 3799

for example, to factor 72257, first check whether it is divisible by 2, 3, 5, or 7.  it is not.  divide by 210 to yield quotient 344 and remainder 17.  go to row 344 and the column headed by "17" to find "d", which corresponds to prime 19.  19 divides 72257 evenly, yielding 3803.  divide 3803 by 210 to yield quotient 18 and remainder 23.  go to row 18, column "23" to find a period, so 3803 is prime.  thus, 72257 = 19 * 3803 .

when the smallest factor is larger than 263 ("Z"), the entry is marked with an asterisk and the factor is given to the right, in the format Column:Factor.  in the last line, 344*210 + 121 = 72361 is divisible by 269.

Perl script to produce the table, and extend it.

future work: use diacritics to extend the range of symbols.  because we are using a fixed width font, accents can go on undotted i and j without interfering with neighboring characters.  also consider Øø (which can take diacritics), which could be collated with after O (making the Danes and Norwegians unhappy) or after Z.  also consider adding the subset of Greek lowercase (and lowercase with ascenders, which cannot take diacritics) and uppercase letters that look different from Latin.  the Greek alphabet has a canonical order (but how many people know the order?), so letters can be found by binary search.

[awtmshvz] higher dimensional cyclic cellular automata

cyclic cellular automata converge to spirals in 2D.  what happens in higher dimensions?  this should be easy.

visualize 3D by making all but one of the colors (states) invisible and transparent.  it will probably start looking like dust randomly moving about, but later perhaps evolve into shifting lines and planes.  the one remaining color should be translucent to be able to see internal structures should anything interesting evolve.

there's a problem in that all objects in the world will be the same color, probably making it difficult to distinguish foreground from background.  maybe things in the distance fade to black.

or, examine 2D cross sections.

[yokcmzvw] partitioning into random shaped rooms

pick a random cell on a grid.  if the cell is already part of a region, pick another cell.  if the cell is adjacent to a region, assign it to that region.  if the cell is adjacent to two or more regions, assign it to the smallest region, picking one at random if a tie for smallest size.  if the cell is adjacent to no regions, it is the seed for a new region.  repeat until there are no unassigned cells.

what sort of pattern of regions results?  what is the distribution of region areas?  what is the distribution of number of neighbors?

it is possible for a region to be completely surrounded by another region.

run a maze generation algorithm on the graph of adjacencies between regions: some region boundaries get doorways.  (aesthetically, where on a boundary should a door go?)  this results in a maze of rooms, with each room having a distinct shape, which can make a maze more entertaining.

further operations: regions merging, dividing.  a region might transfer a border cell to an adjacent region, modeling warfare.

[tdjyidmd] HDR to dithered True Color

convert a high dynamic range (more than 24-bit color) image down to 24-bit color using dithering.

it does not seem straightforward to do with Netpbm.  it seems like a good feature that could be added to pamdepth.  maybe it's possible by providing a colormap with all 2^24 colors to pnmremap.

if one is both downscaling in resolution and reducing color depth, I think the correct order is resolution first, then color depth.

[idmqfwon] clunder

clunder: a classic blunder, a blunder of historical importance.

never get involved in a land war in Asia.

invading Russia in the winter time?

[vgkevtpx] sampling large binomial with random-fu

here are some results of single samples from the binomial distribution using Data.Random.Distribution.Binomial in the random-fu Haskell package, version 0.2.7.0 .

N=320000000000000000000000000000000 p=0.5 sample=160000000000000017763568049748834

N=330000000000000000000000000000000 p=0.5 sample=165000000000000000000000000000000

N=320000000000000000000000000000000 p=6.25e-2 sample=20000000000000004440892003541839

N=330000000000000000000000000000000 p=6.25e-2 sample=20625000000000000000000000000000

Haskell source code.

in the first line, our one sample from the binomial distribution is equivalent to simulating flipping a fair coin 3.2*10^32 times and counting the number of heads.  (however, all these results are calculated instantly.)  things seem to work correctly: about half heads, with statistical noise of order sqrt(N) as would be expected from a random sample.

in the second line, we flip 3.3*10^32 times.  things have gone wrong: the number of heads is precisely half with no statistical noise.

the following two examples use an unfair coin whose probability of success is 1/16.  (we use probabilities expressible exactly in binary to eliminate decimal-to-binary conversion as a possible source of noise).  things go wrong similarly: the threshold does not depend on the probability.

the threshold seems to be around N=2^108, probably related to 53 bits of mantissa in double precision (108 / 2 = 54).  (incidentally, 2^108 ~= 3.2*10^32.  the digits of the decimal mantissa coincide with the exponent.  previously similar.)

this issue is mentioned in comments in the source code of integralBinomial random-fu, but not in the Haddock documentation.  (that Hackage makes it easy to browse source code makes it much better than similar library documentation for other programming languages.  one wonders if the Log4j/Log4shell vulnerability would have been discovered sooner if Java documentation had made it easier to view implementation source code.)

on one hand, of course it's unfortunate that things go wrong for large N.  on the other hand, the failure mode is not too bad.  (there might not be much incentive to fix the bug.)  if you're going to sample the binomial, getting the mode (the most likely result) as an answer might be fine for many applications.

inspired by wanting to draw lots and lots of stars.  if one always gets the mode, there will not be pixels containing a statistically unusual large number of stars.

sampling the binomial distribution efficiently for large N is a non-trivial but seemingly well studied topic.  the comment in random-fu's source code cites Knuth's TAOCP.  wikipedia cites:

Devroye, Luc (1986) Non-Uniform Random Variate Generation, New York: Springer-Verlag.

and links to an online chapter of the book.

[sjosawel] electroweak epoch nostalgia

you kids these days with your distinct electromagnetic and weak nuclear forces.  in my day, we only had the electroweak force, and we liked it.

[wipavyvt] flour dog

replace the cornbread around a corn dog with a different quick bread.  surely this has been done already, perhaps by people who don't like cornbread.  is there a name for this food?

maybe pigs in a blanket, klobasnek, sausage roll.

[alzyzzle] 54-40 and Alaska

if the 54-40 or Fight proponents had gotten their way, and the U.S. purchase of Alaska also remained the same, then Canada would have no access to the Pacific.

Pacific northwest Inside Passage, North American Pacific fjordland.

inspired by Vancouver being Canada's primary port for trade with Asia, which became problematic due to floods in British Columbia cutting Vancouver off from the rest of Canada.  ground transport from Vancouver to the rest of Canada had to route through the U.S.  the Port of Prince Rupert may have also been used instead of Vancouver.

[vbffnrxd] continued fraction expansion of truncated sum of prime reciprocals

length after 10000 primes, Mathematica:

Length[ContinuedFraction[Total[1/Prime[Range[10000]]]]]

88444

(note: the sum diverges, very slowly, when taken over all primes.)

number of continued fraction terms grows faster then linearly.  conjecture n*log(n).

FromContinuedFraction can check that the sum matches.

until termination, the terms don't look unusual, so this can create a rational number that initially looks like a typical irrational number (Khinchin's constant).  (but sampling a finite number of random digits past the decimal point probably also works just as well.)

ContinuedFraction[Total[1/Prime[Range[100]]]]

{2, 9, 2, 2, 10, 1, 2, 12, 14, 1, 8, 1, 1, 18, 1, 1, 5, 1, 1, 5, 5, 1, 4, 1, 10, 1, 3, 3, 2, 24, 2, 1, 3, 1, 5, 6, 5, 1, 58, 1, 1, 2, 1, 2, 6, 3, 1, 2, 3, 1, 1, 2, 3, 1, 34, 1, 2, 1, 2, 2, 1, 19, 1, 7, 1, 16, 7, 14, 1, 1, 1, 18, 1, 2, 6, 1, 1, 1, 2, 6, 6, 3, 1, 4, 34, 408, 1, 1, 1, 87, 85, 1, 45, 1, 1, 3, 12, 1, 2, 3, 2, 1, 4, 6, 27, 1, 1, 6, 1, 1, 539, 2, 2, 1, 2, 1, 1, 1, 3, 3, 192, 10, 167, 2, 6, 7, 2, 1, 4, 1, 1, 1, 1, 1, 3, 13, 2, 5, 1, 10, 2, 2, 3, 1, 119, 14, 2, 1, 3, 1, 2, 1, 1, 3, 11, 13, 2, 4, 3, 13, 165, 1, 4, 1, 1, 1, 3, 1, 7, 1, 5, 3, 15, 1, 1, 181, 1, 1, 3, 1, 1, 9, 1, 1, 4, 3, 1, 1, 4, 1, 2, 6, 2, 2, 11, 1, 4, 5, 15, 3, 1, 6, 2, 9, 1, 1, 1, 5, 14, 3, 2, 1, 1, 1, 1, 2, 1, 2, 126, 1, 2, 1, 1, 1, 3, 1, 82, 8, 1, 1, 11, 24, 4, 1, 2, 1, 3, 19, 2, 1, 1, 1, 1, 1, 29, 1, 2, 3, 1, 1, 9, 2, 7, 1, 3, 1, 4, 1, 1, 2, 24, 1, 6, 6, 2, 4, 9, 1, 3, 1, 1, 1, 42, 1, 20, 2, 2, 1, 1, 1, 2, 6, 1, 4, 1, 22, 42, 1, 142, 2, 4, 2, 1, 5, 39, 1, 5, 1, 5, 1, 9, 1, 5, 1, 1, 4, 1, 13, 26, 1, 1, 51, 1, 58, 1, 1, 4, 9, 14, 1, 7, 20, 2, 1, 2, 1, 1, 1, 3, 1, 6, 9, 1, 1, 1, 67, 4, 2, 1, 1, 3, 3, 1, 4, 2, 3, 2, 1, 1, 6, 3, 1, 10, 2, 1, 8, 1, 1, 1, 1, 99, 70, 2, 1, 1, 1, 4, 1, 10, 9, 5, 1, 1, 1, 1, 1, 3, 36, 1, 19, 69, 5, 110, 1, 9, 1, 1, 2, 2, 1, 15, 2, 5, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 2, 3, 3}

407 terms.

[otodgekv] printf vs cout speed

we demonstrate printf performing over 10x faster than C++ iostreams cout.  this test program prints a bunch of intergers:

#include <cstdlib>
#include <cstdio>

#include <iostream>
using std::endl;
using std::cerr;

void testc(int n){
  for(int i=0;i<n;++i){
    printf("%d\n",i);
  }
  printf("completed printf\n");
}

void testcpp(int n){
  for(int i=0;i<n;++i){
    std::cout << i << endl;
  }
  std::cout << "completed cout" << endl;
}

int main(int argc,char**argv){
  if(argc<3){
    cerr << "args: n choice" << endl;
    return 1;
  }
  // 50 million is a good n
  int n=atoi(argv[1]);

  int choice=atoi(argv[2]);
  if(0==choice)
    testc(n);
  else if(1==choice)
    testcpp(n);
  else {
    cerr << "bad choice" << endl;
  }
}

$ g++ -v
...
gcc version 7.5.0 (Ubuntu 7.5.0-3ubuntu1~18.04)

Test printf:

$ command time ./a.out 50000000 0 | tail -2
4.04user 0.67system 0:04.71elapsed 99%CPU (0avgtext+0avgdata 3488maxresident)k
0inputs+0outputs (0major+125minor)pagefaults 0swaps
49999999
completed printf

$ command time ./a.out 50000000 0 > /dev/null
4.04user 0.05system 0:04.10elapsed 100%CPU (0avgtext+0avgdata 3456maxresident)k
0inputs+0outputs (0major+124minor)pagefaults 0swaps

Test iostreams cout:

$ command time ./a.out 50000000 1 | tail -2
25.52user 41.49system 1:07.03elapsed 99%CPU (0avgtext+0avgdata 3392maxresident)k
0inputs+0outputs (0major+123minor)pagefaults 0swaps
49999999
completed cout

$ command time ./a.out 50000000 1 > /dev/null
23.86user 22.85system 0:46.71elapsed 100%CPU (0avgtext+0avgdata 3348maxresident)k
0inputs+8outputs (0major+124minor)pagefaults 0swaps

when piping to tail -2, elapsed time slowdown factor was 14.2 .  when redirecting to /dev/null , elapsed time slowdown factor was 11.4 .

motivation for printing a bunch of integers was emitting Netpbm's easy-to-generate P3 text-only "noraw" PPM image format, then piping to pnmtopng .

on a different machine and OS (Debian Buster) and gcc version 8.3.0 (Debian 8.3.0-6), still saw significant but not quite as dramatic slowdowns.  when piping to tail -2, elapsed time slowdown factor was 9.9 .  when redirecting to /dev/null , elapsed time slowdown factor was 6.4 .

with gcc version 10.2.1 20210110 (Debian 10.2.1-6), Debian Bullseye, on Xen Dom0 under constant high load: slowdown factors 48.4 and 19.2 .

the classic "Hello, world!" task is rather nontrivial.

[qvqmyhei] surprisingly rational or algebraic

given no prior information about a number, it's probably an irrational number, probably a transcendental number.  what are some exceptions to this rule of thumb?

geometric series

growth rate of look-and-say sequence

exp(i*pi)

[nqmtuzvi] macrowave

what would it be like to cook with a microwave oven much more powerful than a normal consumer device?

using an electric stove power outlet, it would probably not be too difficult to draw 3x power.

can exhaust heat (from cooling microwave electronics) be harnessed?  microwave ovens are supposedly about 50% efficient.  maybe for warming water, always useful for cooking.

[uizrizee] orbits partially explained

the fact that it is possible for a satellite to orbit the earth is unintuitive: what force keeps the satellite suspended in the sky?

"what goes up must come down" is good intuition if the earth is flat.  but the earth isn't flat: the direction of "down" changes depending on where you are.  if a projectile is moving horizontally so fast that its direction of "down" changes significantly over the course of its trajectory, then one can imagine that something more complicated than "what goes up must come down" might occur, and it does.

determinating exactly what does occur, for example, showing that a circular orbit is possible, requires more work, probably some actual math.  (actually, circular orbits are not possible with general relativity because of gravitational radiation.  an orbiting satellite therefore does not stay up forever, just as intuition originally suggested.  but calculating the inspiral requires a supercomputer.)

[geaapvru] Area 51 black site

patriots storm Area 51 and, against all odds, overrun the facility.  they discover not imprisoned aliens but imprisoned humans: the top secret government facility with no public access is a torture site.  in retrospect, this comes as a surprise to no one.

is there demand for torture on US soil?  it seems yes: a problem with offshoring torture to other countries is that they too get to learn the extracted information.

how should a torture site on U.S. soil be set up?  it needs to have a cover story for plausible deniability.  it needs to have an airstrip so that prisoners can be transported discreetly and without potential obstruction or interruption.  it needs to be in a remote location so that outsiders cannot hear the screams or smell the rotting or burning flesh.  Area 51 satisfies these requirements.

Guantanamo Bay (Gitmo) seems not a good site because there are too many soldiers around without high security clearance who might witness evidence of torture happening, then do their constitutional duty and leak.

however, it seems a bad idea to colocate a torture site with one doing actual secret military technology development, because one needs to vet employees, soldiers, to be the kind of people who will keep mum about both military technology and violations of the Geneva Conventions.  that overlap might not be very large: techies in the former category, fuzzies in the latter.

[pnzmtbew] avoiding CMY brighter than RGB

consider traveling around the saturated colors in the color wheel (future post gxmkhjal).  on an RGB display, cyan, magenta, and yellow will (probably) be brighter than their neighbors on the wheel red, green, and blue because more pixels (subpixels) are turned on.

instead, compute not fully saturated values of C, M, and Y so that the brightness of the color ramps between saturated red, green, and blue points is linear.

this should be easy.  easiest is to use the Y component of YCbCr (which is slightly different from YUV); also possible is Cielab.

gamma will make things complicated.

[yhvjfdmq] faster speed of light

two science-fictional universes:

speed of light is infinite.  fun with world building: invent completely different theories of electromagnetism, special relativity, and gravity.

speed of light is finite but a number too large to comprehend, similar to Graham's number.  how was it discovered to be finite?  this scenario could happen in our universe with slow intelligent life.

in both, high speed interstellar travel is possible without breaking special relativity, as is instantaneous or nearly instantaneous communication between distant worlds.

[jlnjchpr] bisecting trees

given a graph that is a tree, find an edge that, if cut, divides the tree into as equal sized pair of pieces as possible.  (there may be multiple possible such cuts; choose one.)  repeat for the subtrees.

motivation is to place some helpful landmarks (entrances to "regions") in a tree-like maze.  do the first few layers of landmarks end up all being near each other?

[enqwdtqg] Japanese R/L

English-language comedy often makes fun of a Japanese character being unable to pronounce the difference between L and R.  is there corresponding Japanese comedy about an English speaker being unable to pronounce the halfway-between-L-and-R consonant of Japanese?

"one American-style ramen coming right up for you.  it's fravored with ketchup and bacon, so you will rove it."

[cyutoegx] formal proof of simplify

when a computer algebra program (e.g., Mathematica) simplifies an expression, let it optionally emit a formal proof of the simplification which can be checked by an independent proof assistant program.  the formal proof may be ugly and messy; it is intended only to be processed by a machine.

make formal methods widely used and available.

[smnvmcqw] transmuting the earth

consider transmuting every atom on earth into a given element via nuclear fusion or fission.  which target elements require net energy?  which release net energy?  earth has a lot of iron, so transmuting to most elements will probably require energy.

how do the energies compare to the gravitational binding energy of the earth?  previously.

these should be easy to compute: look up nuclear binding energies.  we need to know the elemental and isotopic composition of the earth, which surprisingly is known despite us never having taken a core sample.  probably estimated from meteorites, though I don't know the details.

[lqqqdyqg] password hashing in Perl

here is a simple Perl script which hashes a password with 300000 rounds of key stretching using crypt(3).

previously, a fancier script.

[mhuywhmr] pandiagonal magic squares

pandiagonal magic squares are interesting because they have a lot of internal error correction: values can be altered or erased but the correct square can usually be recovered.  (is this true?)

what is an algorithm to randomly generate a pandiagonal magic square?  ideally uniformly sample from all possible pandiagonal magic squares, but that seems difficult, so sampling from a substantial subset is good enough.

consider robustly encoding data as a collection of pandiagonal magic squares.  each pandiagonal magic square is a glyph.  what is a mapping (bijection) between numbers 1 to N and a collection of pandiagonal magic squares?

[rfbwyohu] Legendre-Gauss quadrature

Gauss-Legendre quadrature seems quite spectacular at first sight, perfectly integrating polynomials of degree 2n-1 after sampling only n points.  however, because the quadrature scheme chooses n abscissae and n weights, on further thought it's not too surprising that it can recover the 2n free parameters of a polynomial of degree 2n-1.

digression: because the scheme only evaluates interior points, it has a fighting chance at working all right when an endpoint is a singularity.  how well does it work in practice?  of course this will depend on the function.

we derive and attempt to verify the quadrature scheme for 5 through 9 points in Mathematica, following https://pomax.github.io/bezierinfo/legendre-gauss.html .

5-point Gauss-Legendre quadrature

the exact expressions of abscissae and weights for 5 points are also published elsewhere.

LegendreP[5,x]

(15*x - 70*x^3 + 63*x^5)/8

solve Legendre polynomial for x.

roots[n_] := x /. Solve[LegendreP[n,x]==0, Cubics->True, Quartics->True]

for n=5, this requires solving a quadratic equation.

roots[5]

{0, -Sqrt[5 - 2*Sqrt[10/7]]/3, Sqrt[5 - 2*Sqrt[10/7]]/3, -Sqrt[5 + 2*Sqrt[10/7]]/3, Sqrt[5 + 2*Sqrt[10/7]]/3}

legendrederivative[n_,a_] := D[LegendreP[n,x],x] /. x->a

legendrederivative[5,x]

(15 - 210*x^2 + 315*x^4)/8

weightfunction[n_,x_] := 2/((1-x^2)*legendrederivative[n,x]^2)

the hash symbol (number sign) and ampersand together create a lambda function, called pure anonymous functions in Mathematica documentation:

weights[n_] := Map[weightfunction[n,#]&,roots[n]]

weights[5]

{128/225, (322 + 13*Sqrt[70])/900, (322 + 13*Sqrt[70])/900, (322 - 13*Sqrt[70])/900, (322 - 13*Sqrt[70])/900}

create a general polynomial of degree 2n-1 :

f[n_,x_] := FromDigits[Array[a,2*n],x]

f[5,x]

x^9*a[1] + x^8*a[2] + x^7*a[3] + x^6*a[4] + x^5*a[5] + x^4*a[6] + x^3*a[7] + x^2*a[8] + x*a[9] + a[10]

here is the correct answer:

Integrate[f[5,x],{x,-1,1}]

(2*a[2])/9 + (2*a[4])/7 + (2*a[6])/5 + (2*a[8])/3 + 2*a[10]

test if quadrature gets the same answer:

g[n_] := Dot[weights[n],Map[f[n,#]&,roots[n]]]

g[5] - Integrate[f[5,x],{x,-1,1}] // Simplify

0

5 point quadrature can exactly integrate a polynomial of degree 9; what happens if we try a 10th degree?

ff[x_] := FromDigits[Array[a,11],x]

ff[x]

x^10*a[1] + x^9*a[2] + x^8*a[3] + x^7*a[4] + x^6*a[5] + x^5*a[6] + x^4*a[7] + x^3*a[8] + x^2*a[9] + x*a[10] + a[11]

Integrate[ff[x],{x,-1,1}]

(2*a[1])/11 + (2*a[3])/9 + (2*a[5])/7 + (2*a[7])/5 + (2*a[9])/3 + 2*a[11]

Dot[weights[5],Map[ff,roots[5]]] -Integrate[ff[x],{x,-1,1}] //Simplify

(-128*a[1])/43659

not only is there error, but there is guaranteed to be error, because a[1] must be nonzero in a 10th degree polynomial.

6-point Legendre-Gauss quadrature

6 and 7 point quadrature require solving a cubic equation, which is possible but messy.

LegendreP[6,x]

(-5 + 105*x^2 - 315*x^4 + 231*x^6)/16

roots[6]

here's what Mathematica says if Cubics->False, Quartics->False :

{-Sqrt[Root[-5 + 105*#1 - 315*#1^2 + 231*#1^3 & , 1, 0]],
Sqrt[Root[-5 + 105*#1 - 315*#1^2 + 231*#1^3 & , 1, 0]],
-Sqrt[Root[-5 + 105*#1 - 315*#1^2 + 231*#1^3 & , 2, 0]],
Sqrt[Root[-5 + 105*#1 - 315*#1^2 + 231*#1^3 & , 2, 0]],
-Sqrt[Root[-5 + 105*#1 - 315*#1^2 + 231*#1^3 & , 3, 0]],
Sqrt[Root[-5 + 105*#1 - 315*#1^2 + 231*#1^3 & , 3, 0]]}

and now Cubics->True, Quartics->True :

{-((-1)^(3/4)*Sqrt[(105*I - ((7*I)*3^(1/6)*5^(2/3)*(-3*I + Sqrt[3]))/ ((3 + (11*I)*Sqrt[6])/7)^(1/3) + 3^(5/6)*7^(2/3)* (15 + (55*I)*Sqrt[6])^(1/3) - I*7^(2/3)*(45 + (165*I)*Sqrt[6])^(1/3))/ 231]), (-1)^(3/4)* Sqrt[(105*I - ((7*I)*3^(1/6)*5^(2/3)*(-3*I + Sqrt[3]))/ ((3 + (11*I)*Sqrt[6])/7)^(1/3) + 3^(5/6)*7^(2/3)* (15 + (55*I)*Sqrt[6])^(1/3) - I*7^(2/3)*(45 + (165*I)*Sqrt[6])^(1/3))/ 231], -(((-1)^(1/4)*Sqrt[((7*I)*3^(1/6)*5^(2/3)*(3*I + Sqrt[3]))/ ((3 + (11*I)*Sqrt[6])/7)^(1/3) + 3^(5/6)*7^(2/3)* (15 + (55*I)*Sqrt[6])^(1/3) + I*(-105 + 7^(2/3)*(45 + (165*I)*Sqrt[6])^(1/3))])/Sqrt[231]), ((-1)^(1/4)*Sqrt[((7*I)*3^(1/6)*5^(2/3)*(3*I + Sqrt[3]))/ ((3 + (11*I)*Sqrt[6])/7)^(1/3) + 3^(5/6)*7^(2/3)* (15 + (55*I)*Sqrt[6])^(1/3) + I*(-105 + 7^(2/3)*(45 + (165*I)*Sqrt[6])^(1/3))])/Sqrt[231], -Sqrt[(105 + (14*15^(2/3))/((3 + (11*I)*Sqrt[6])/7)^(1/3) + 2*7^(2/3)*(45 + (165*I)*Sqrt[6])^(1/3))/231], Sqrt[(105 + (14*15^(2/3))/((3 + (11*I)*Sqrt[6])/7)^(1/3) + 2*7^(2/3)*(45 + (165*I)*Sqrt[6])^(1/3))/231]}

even though we can, doesn't mean we should.  finding roots numerically to any desired precision with Newton's method is likely better practically.  nevertheless, charging onward.

legendrederivative[6,x]

(210*x - 1260*x^3 + 1386*x^5)/16

weights[6]

{(117128*7^(2/3)*(3 + (11*I)*Sqrt[6])^(1/3)*(-3*I + 11*Sqrt[6])^5)/ (135*(7*3^(1/6)*5^(2/3)*(1275547677 - 2235092904*Sqrt[2] + (425182559*I)*Sqrt[3] + (2235092904*I)*Sqrt[6]) + 27970*7^(2/3)*(3 + (11*I)*Sqrt[6])^(1/3)*(-856707*I + 564839*Sqrt[6]) - 35*105^(1/3)*(3 + (11*I)*Sqrt[6])^(2/3)*(-41261489*I + (41781234*I)*Sqrt[2] + 41261489*Sqrt[3] + 13927078*Sqrt[6]))), (117128*7^(2/3)*(3 + (11*I)*Sqrt[6])^(1/3)*(-3*I + 11*Sqrt[6])^5)/ (135*(7*3^(1/6)*5^(2/3)*(1275547677 - 2235092904*Sqrt[2] + (425182559*I)*Sqrt[3] + (2235092904*I)*Sqrt[6]) + 27970*7^(2/3)*(3 + (11*I)*Sqrt[6])^(1/3)*(-856707*I + 564839*Sqrt[6]) - 35*105^(1/3)*(3 + (11*I)*Sqrt[6])^(2/3)*(-41261489*I + (41781234*I)*Sqrt[2] + 41261489*Sqrt[3] + 13927078*Sqrt[6]))), (-117128*7^(2/3)*(3 + (11*I)*Sqrt[6])^(1/3)*(-3*I + 11*Sqrt[6])^5)/ (135*(7*3^(1/6)*5^(2/3)*(1275547677 + 2235092904*Sqrt[2] - (425182559*I)*Sqrt[3] + (2235092904*I)*Sqrt[6]) - 27970*7^(2/3)*(3 + (11*I)*Sqrt[6])^(1/3)*(-856707*I + 564839*Sqrt[6]) + 35*105^(1/3)*(3 + (11*I)*Sqrt[6])^(2/3)*(-41261489*I - (41781234*I)*Sqrt[2] - 41261489*Sqrt[3] + 13927078*Sqrt[6]))), (-117128*7^(2/3)*(3 + (11*I)*Sqrt[6])^(1/3)*(-3*I + 11*Sqrt[6])^5)/ (135*(7*3^(1/6)*5^(2/3)*(1275547677 + 2235092904*Sqrt[2] - (425182559*I)*Sqrt[3] + (2235092904*I)*Sqrt[6]) - 27970*7^(2/3)*(3 + (11*I)*Sqrt[6])^(1/3)*(-856707*I + 564839*Sqrt[6]) + 35*105^(1/3)*(3 + (11*I)*Sqrt[6])^(2/3)*(-41261489*I - (41781234*I)*Sqrt[2] - 41261489*Sqrt[3] + 13927078*Sqrt[6]))), (58564*7^(2/3)*(3 + (11*I)*Sqrt[6])^(1/3)*(-3*I + 11*Sqrt[6])^5)/ (135*(7*3^(1/6)*5^(2/3)*(2235092904*Sqrt[2] - (425182559*I)*Sqrt[3]) + 13985*7^(2/3)*(3 + (11*I)*Sqrt[6])^(1/3)*(-856707*I + 564839*Sqrt[6]) + 35*105^(1/3)*(3 + (11*I)*Sqrt[6])^(2/3)*(-41261489*I + 13927078*Sqrt[6]))), (58564*7^(2/3)*(3 + (11*I)*Sqrt[6])^(1/3)*(-3*I + 11*Sqrt[6])^5)/ (135*(7*3^(1/6)*5^(2/3)*(2235092904*Sqrt[2] - (425182559*I)*Sqrt[3]) + 13985*7^(2/3)*(3 + (11*I)*Sqrt[6])^(1/3)*(-856707*I + 564839*Sqrt[6]) + 35*105^(1/3)*(3 + (11*I)*Sqrt[6])^(2/3)*(-41261489*I + 13927078*Sqrt[6])))}

the weights above are real numbers.  the presence of the imaginary unit I is casus irreducibilis.

f[6,x]

x^11*a[1] + x^10*a[2] + x^9*a[3] + x^8*a[4] + x^7*a[5] + x^6*a[6] + x^5*a[7] + x^4*a[8] + x^3*a[9] + x^2*a[10] + x*a[11] + a[12]

Integrate[f[6,x],{x,-1,1}]

(2*a[2])/11 + (2*a[4])/9 + (2*a[6])/7 + (2*a[8])/5 + (2*a[10])/3 + 2*a[12]

Mathematica creaks and moans but is able to prove equality.

Timing[Simplify[g[6] - Integrate[f[6,x],{x,-1,1}]]]

Simplify::time: Time spent on a transformation exceeded 300. seconds, and the transformation was aborted. Increasing the value of TimeConstraint option may improve the result of simplification.

(326.626 seconds)

0

Timing[FullSimplify[g[6] == Integrate[f[6,x],{x,-1,1}]]]//InputForm

(1845 seconds, with or without Cubics->True, Quartics->True)

True

7-point Legendre-Gauss quadrature

LegendreP[7,x]

(-35*x + 315*x^3 - 693*x^5 + 429*x^7)/16

roots[7]

Cubics->False, Quartics->False

{0,
-Sqrt[Root[-35 + 315*#1 - 693*#1^2 + 429*#1^3 & , 1, 0]],
Sqrt[Root[-35 + 315*#1 - 693*#1^2 + 429*#1^3 & , 1, 0]],
-Sqrt[Root[-35 + 315*#1 - 693*#1^2 + 429*#1^3 & , 2, 0]],
Sqrt[Root[-35 + 315*#1 - 693*#1^2 + 429*#1^3 & , 2, 0]],
-Sqrt[Root[-35 + 315*#1 - 693*#1^2 + 429*#1^3 & , 3, 0]],
Sqrt[Root[-35 + 315*#1 - 693*#1^2 + 429*#1^3 & , 3, 0]]}

Cubics->True, Quartics->True

{0, -((-1)^(3/4)*Sqrt[(9*3^(1/3)*77^(2/3)*(-I + Sqrt[3]) + (231*I)*(-11 + (13*I)*Sqrt[110])^(1/3) - 3*3^(1/6)*77^(1/3)* (-11 + (13*I)*Sqrt[110])^(2/3) - I*77^(1/3)*(-33 + (39*I)*Sqrt[110])^ (2/3))/(429*(-11 + (13*I)*Sqrt[110])^(1/3))]), (-1)^(3/4)*Sqrt[(9*3^(1/3)*77^(2/3)*(-I + Sqrt[3]) + (231*I)*(-11 + (13*I)*Sqrt[110])^(1/3) - 3*3^(1/6)*77^(1/3)* (-11 + (13*I)*Sqrt[110])^(2/3) - I*77^(1/3)*(-33 + (39*I)*Sqrt[110])^ (2/3))/(429*(-11 + (13*I)*Sqrt[110])^(1/3))], -(((-1)^(1/4)*Sqrt[-231*I + (9*77^(2/3)*(I + Sqrt[3]))/ ((-11 + (13*I)*Sqrt[110])/3)^(1/3) - 3*3^(1/6)*(77*(-11 + (13*I)*Sqrt[110]))^(1/3) + I*3^(2/3)*(77*(-11 + (13*I)*Sqrt[110]))^(1/3)])/Sqrt[429]), ((-1)^(1/4)*Sqrt[-231*I + (9*77^(2/3)*(I + Sqrt[3]))/ ((-11 + (13*I)*Sqrt[110])/3)^(1/3) - 3*3^(1/6)*(77*(-11 + (13*I)*Sqrt[110]))^(1/3) + I*3^(2/3)*(77*(-11 + (13*I)*Sqrt[110]))^(1/3)])/Sqrt[429], -Sqrt[(231 + (18*77^(2/3))/((-11 + (13*I)*Sqrt[110])/3)^(1/3) + 2*3^(2/3)*(77*(-11 + (13*I)*Sqrt[110]))^(1/3))/429], Sqrt[(231 + (18*77^(2/3))/((-11 + (13*I)*Sqrt[110])/3)^(1/3) + 2*3^(2/3)*(77*(-11 + (13*I)*Sqrt[110]))^(1/3))/429]}

legendrederivative[7,x]

(-35 + 945*x^2 - 3465*x^4 + 3003*x^6)/16

weights[7]

{512/1225, (359411624*(-11 + (13*I)*Sqrt[110])^(1/3)*(-1679*I + 26*Sqrt[110]))/ (147*(3*3^(1/6)*77^(1/3)*(-11 + (13*I)*Sqrt[110])^(2/3)* (69836087 - (1435135*I)*Sqrt[110]) + 11342298*(-11 + (13*I)*Sqrt[110])^(1/3)*(-1679*I + 26*Sqrt[110]) + 11^(1/6)*21^(1/3)*((-3*I)*7^(1/3)*(-I + Sqrt[3])*(114465923*Sqrt[10] - (184448941*I)*Sqrt[11]) + (69836087*I)*3^(1/3)*11^(1/6)* (-11 + (13*I)*Sqrt[110])^(2/3) + 1435135*3^(1/3)*Sqrt[10]* (-121 + (143*I)*Sqrt[110])^(2/3)))), (359411624*(-11 + (13*I)*Sqrt[110])^(1/3)*(-1679*I + 26*Sqrt[110]))/ (147*(3*3^(1/6)*77^(1/3)*(-11 + (13*I)*Sqrt[110])^(2/3)* (69836087 - (1435135*I)*Sqrt[110]) + 11342298*(-11 + (13*I)*Sqrt[110])^(1/3)*(-1679*I + 26*Sqrt[110]) + 11^(1/6)*21^(1/3)*((-3*I)*7^(1/3)*(-I + Sqrt[3])*(114465923*Sqrt[10] - (184448941*I)*Sqrt[11]) + (69836087*I)*3^(1/3)*11^(1/6)* (-11 + (13*I)*Sqrt[110])^(2/3) + 1435135*3^(1/3)*Sqrt[10]* (-121 + (143*I)*Sqrt[110])^(2/3)))), (359411624*(-11 + (13*I)*Sqrt[110])^(1/3)*(-1679*I + 26*Sqrt[110]))/ (147*(3*3^(1/6)*77^(1/3)*(-11 + (13*I)*Sqrt[110])^(2/3)* (-69836087 + (1435135*I)*Sqrt[110]) + 11342298*(-11 + (13*I)*Sqrt[110])^(1/3)*(-1679*I + 26*Sqrt[110]) + 11^(1/6)*21^(1/3)*(3*7^(1/3)*(I + Sqrt[3])*((114465923*I)*Sqrt[10] + 184448941*Sqrt[11]) + (69836087*I)*3^(1/3)*11^(1/6)* (-11 + (13*I)*Sqrt[110])^(2/3) + 1435135*3^(1/3)*Sqrt[10]* (-121 + (143*I)*Sqrt[110])^(2/3)))), (359411624*(-11 + (13*I)*Sqrt[110])^(1/3)*(-1679*I + 26*Sqrt[110]))/ (147*(3*3^(1/6)*77^(1/3)*(-11 + (13*I)*Sqrt[110])^(2/3)* (-69836087 + (1435135*I)*Sqrt[110]) + 11342298*(-11 + (13*I)*Sqrt[110])^(1/3)*(-1679*I + 26*Sqrt[110]) + 11^(1/6)*21^(1/3)*(3*7^(1/3)*(I + Sqrt[3])*((114465923*I)*Sqrt[10] + 184448941*Sqrt[11]) + (69836087*I)*3^(1/3)*11^(1/6)* (-11 + (13*I)*Sqrt[110])^(2/3) + 1435135*3^(1/3)*Sqrt[10]* (-121 + (143*I)*Sqrt[110])^(2/3)))), (((16336892*I)/147)*(-11 + (13*I)*Sqrt[110])^(1/3)*(11*I + 13*Sqrt[110])^2)/ (-5671149*(-11 + (13*I)*Sqrt[110])^(1/3)*(-1679*I + 26*Sqrt[110]) + 11^(1/6)*21^(1/3)*(3*7^(1/3)*(-114465923*Sqrt[10] + (184448941*I)*Sqrt[11]) + (69836087*I)*3^(1/3)*11^(1/6)* (-11 + (13*I)*Sqrt[110])^(2/3) + 1435135*3^(1/3)*Sqrt[10]* (-121 + (143*I)*Sqrt[110])^(2/3))), (((16336892*I)/147)*(-11 + (13*I)*Sqrt[110])^(1/3)*(11*I + 13*Sqrt[110])^2)/ (-5671149*(-11 + (13*I)*Sqrt[110])^(1/3)*(-1679*I + 26*Sqrt[110]) + 11^(1/6)*21^(1/3)*(3*7^(1/3)*(-114465923*Sqrt[10] + (184448941*I)*Sqrt[11]) + (69836087*I)*3^(1/3)*11^(1/6)* (-11 + (13*I)*Sqrt[110])^(2/3) + 1435135*3^(1/3)*Sqrt[10]* (-121 + (143*I)*Sqrt[110])^(2/3)))}

f[7,x]

x^13*a[1] + x^12*a[2] + x^11*a[3] + x^10*a[4] + x^9*a[5] + x^8*a[6] + x^7*a[7] + x^6*a[8] + x^5*a[9] + x^4*a[10] + x^3*a[11] + x^2*a[12] + x*a[13] + a[14]

Integrate[f[7,x],{x,-1,1}]

(2*a[2])/13 + (2*a[4])/11 + (2*a[6])/9 + (2*a[8])/7 + (2*a[10])/5 + (2*a[12])/3 + 2*a[14]

curiously, proving correctness of 7 points is faster than 6 points.

Timing[Simplify[g[7] - Integrate[f[7,x],{x,-1,1}]]]

(6.78219 seconds)

0

8-point Legendre-Gauss quadrature

LegendreP[8,x]

(35 - 1260*x^2 + 6930*x^4 - 12012*x^6 + 6435*x^8)/128

roots[8]

Cubics->False, Quartics->False

{-Sqrt[Root[35 - 1260*#1 + 6930*#1^2 - 12012*#1^3 + 6435*#1^4 & , 1, 0]],
Sqrt[Root[35 - 1260*#1 + 6930*#1^2 - 12012*#1^3 + 6435*#1^4 & , 1, 0]],
-Sqrt[Root[35 - 1260*#1 + 6930*#1^2 - 12012*#1^3 + 6435*#1^4 & , 2, 0]],
Sqrt[Root[35 - 1260*#1 + 6930*#1^2 - 12012*#1^3 + 6435*#1^4 & , 2, 0]],
-Sqrt[Root[35 - 1260*#1 + 6930*#1^2 - 12012*#1^3 + 6435*#1^4 & , 3, 0]],
Sqrt[Root[35 - 1260*#1 + 6930*#1^2 - 12012*#1^3 + 6435*#1^4 & , 3, 0]],
-Sqrt[Root[35 - 1260*#1 + 6930*#1^2 - 12012*#1^3 + 6435*#1^4 & , 4, 0]],
Sqrt[Root[35 - 1260*#1 + 6930*#1^2 - 12012*#1^3 + 6435*#1^4 & , 4, 0]]}

Cubics->True, Quartics->True

{-Sqrt[7/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2], Sqrt[7/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2], -Sqrt[7/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2], Sqrt[7/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2], -Sqrt[7/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2], Sqrt[7/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2], -Sqrt[7/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2], Sqrt[7/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2]}

legendrederivative[8,x]

(-2520*x + 27720*x^3 - 72072*x^5 + 51480*x^7)/128

weights[8]

(151.66717 seconds)

{32768/((8/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)* (2520*Sqrt[7/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2] - 27720*(7/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(3/2) + 72072*(7/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(5/2) - 51480*(7/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(7/2))^2), 32768/((8/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)* (2520*Sqrt[7/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2] - 27720*(7/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(3/2) + 72072*(7/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(5/2) - 51480*(7/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(7/2))^2), 32768/((8/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)* (2520*Sqrt[7/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2] - 27720*(7/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(3/2) + 72072*(7/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(5/2) - 51480*(7/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(7/2))^2), 32768/((8/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)* (2520*Sqrt[7/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2] - 27720*(7/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(3/2) + 72072*(7/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(5/2) - 51480*(7/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) - 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(7/2))^2), 32768/((8/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)* (2520*Sqrt[7/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2] - 27720*(7/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(3/2) + 72072*(7/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(5/2) - 51480*(7/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(7/2))^2), 32768/((8/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)* (2520*Sqrt[7/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2] - 27720*(7/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(3/2) + 72072*(7/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(5/2) - 51480*(7/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(7/2))^2), 32768/((8/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)* (2520*Sqrt[7/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2] - 27720*(7/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(3/2) + 72072*(7/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(5/2) - 51480*(7/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(7/2))^2), 32768/((8/15 - (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 - Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)* (2520*Sqrt[7/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2] - 27720*(7/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(3/2) + 72072*(7/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(5/2) - 51480*(7/15 + (2*Sqrt[(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))/143])/15 + Sqrt[896/2925 - (16*14^(2/3))/(39*(165*(42 + I*Sqrt[546]))^(1/3)) - (16*(588 + (14*I)*Sqrt[546])^(1/3))/(39*165^(2/3)) + 896/(225*Sqrt[143*(308 + (5*2310^(2/3))/(42 + I*Sqrt[546])^(1/3) + 5*(2310*(42 + I*Sqrt[546]))^(1/3))])]/2)^(7/2))^2)}

f[8,x]

x^15*a[1] + x^14*a[2] + x^13*a[3] + x^12*a[4] + x^11*a[5] + x^10*a[6] + x^9*a[7] + x^8*a[8] + x^7*a[9] + x^6*a[10] + x^5*a[11] + x^4*a[12] + x^3*a[13] + x^2*a[14] + x*a[15] + a[16]

Integrate[f[8,x],{x,-1,1}]

(2*a[2])/15 + (2*a[4])/13 + (2*a[6])/11 + (2*a[8])/9 + (2*a[10])/7 + (2*a[12])/5 + (2*a[14])/3 + 2*a[16]

Timing[Simplify[g[8] - Integrate[f[8,x],{x,-1,1}], TimeConstraint->14400]]

without the TimeConstraint, Mathematica fails to Simplify to zero.  with the TimeConstraint, Mathematica gives up, mysteriously exiting without an error or status code.  Mathematica also mysteriously gives up on the following:

Timing[FullSimplify[g[8] == Integrate[f[8,x],{x,-1,1}]]]

trying again without specifying Cubics->True, Quartics->True

roots[n_] := x /. Solve[LegendreP[n,x]==0]

then we do succeed, though it's mysterious why.  how do you do algebra on Root[] expressions?

Timing[FullSimplify[g[8] == Integrate[f[8,x],{x,-1,1}]]]

{254.274571, True}

we also demonstrate quadrature working on a polynomial carefully designed to have a pretty answer:

nf[n_,y_]:=D[7*(x^(2*n+1)-1)/(x-1) /. x-> (x+1)*5,x] /. x -> y

nf[8,x]//Factor//Expand

21028518676760 + 310587882995600*x + 2141341567039500*x^2 + 9141668677330000*x^3 + 27025499381125000*x^4 + 58604206983000000*x^5 + 96296700916718750*x^6 + 122092097881250000*x^7 + 120424152515625000*x^8 + 92403046093750000*x^9 + 54706981054687500*x^10 + 24542027343750000*x^11 + 8075361328125000*x^12 + 1839892578125000*x^13 + 259552001953125*x^14 + 17089843750000*x^15

Integrate[nf[8,x],{x,-1,1}]

77777777777777770

gn[n_]:=Dot[weights[n],Map[nf[n,#]&,roots[n]]]

N[gn[8],25] //InputForm

7.777777777777777`25.15051499783199*^16 + 0``8.259659467257055*I

N[Integrate[nf[8,x],{x,-1,1}],25] //InputForm

7.777777777777777`25.*^16

9-point Legendre-Gauss quadrature

LegendreP[9,x]

(315*x - 4620*x^3 + 18018*x^5 - 25740*x^7 + 12155*x^9)/128

roots[9]

Cubics->False, Quartics->False

{0,
-Sqrt[Root[315 - 4620*#1 + 18018*#1^2 - 25740*#1^3 + 12155*#1^4 & , 1, 0]],
Sqrt[Root[315 - 4620*#1 + 18018*#1^2 - 25740*#1^3 + 12155*#1^4 & , 1, 0]],
-Sqrt[Root[315 - 4620*#1 + 18018*#1^2 - 25740*#1^3 + 12155*#1^4 & , 2, 0]],
Sqrt[Root[315 - 4620*#1 + 18018*#1^2 - 25740*#1^3 + 12155*#1^4 & , 2, 0]],
-Sqrt[Root[315 - 4620*#1 + 18018*#1^2 - 25740*#1^3 + 12155*#1^4 & , 3, 0]],
Sqrt[Root[315 - 4620*#1 + 18018*#1^2 - 25740*#1^3 + 12155*#1^4 & , 3, 0]],
-Sqrt[Root[315 - 4620*#1 + 18018*#1^2 - 25740*#1^3 + 12155*#1^4 & , 4, 0]],
Sqrt[Root[315 - 4620*#1 + 18018*#1^2 - 25740*#1^3 + 12155*#1^4 & , 4, 0]]}

Cubics->True, Quartics->True

{0, -Sqrt[9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 - Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/2], Sqrt[9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 - Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/2], -Sqrt[9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/2], Sqrt[9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/2], -Sqrt[9/17 - (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 - Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/2], Sqrt[9/17 - (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 - Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/2], -Sqrt[9/17 - (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/2], Sqrt[9/17 - (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/2]}

legendrederivative[9,x]

(315 - 13860*x^2 + 90090*x^4 - 180180*x^6 + 109395*x^8)/128

weights[9]

(90.535658 seconds)

{32768/99225, 32768/ ((8/17 - (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/2)* (315 - 13860*(9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/ 17 - Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ (85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^ (1/3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^ (1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^ (1/3)])]/2) + 90090*(9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^ (2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/ 17 - Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ (85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/ 3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^2 - 180180* (9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 - Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^3 + 109395* (9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 - Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^4)^2), 32768/ ((8/17 - (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/2)* (315 - 13860*(9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/ 17 - Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ (85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^ (1/3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^ (1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^ (1/3)])]/2) + 90090*(9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^ (2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/ 17 - Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ (85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/ 3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^2 - 180180* (9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 - Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^3 + 109395* (9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 - Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^4)^2), 32768/ ((8/17 - (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 - Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/2)* (315 - 13860*(9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/ 17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ (85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^ (1/3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^ (1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^ (1/3)])]/2) + 90090*(9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^ (2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/ 17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ (85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/ 3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^2 - 180180* (9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^3 + 109395* (9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^4)^2), 32768/ ((8/17 - (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 - Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/2)* (315 - 13860*(9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/ 17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ (85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^ (1/3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^ (1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^ (1/3)])]/2) + 90090*(9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^ (2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/ 17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ (85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/ 3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^2 - 180180* (9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^3 + 109395* (9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) - 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^4)^2), 32768/ ((8/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/2)* (315 - 13860*(9/17 - (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/ 17 - Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ (85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^ (1/3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^ (1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^ (1/3)])]/2) - 180180*(9/17 - (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^ (2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/ 17 - Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ (85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/ 3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^3 + 90090* (-9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^2 + 109395* (-9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^4)^2), 32768/ ((8/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/2)* (315 - 13860*(9/17 - (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/ 17 - Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ (85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^ (1/3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^ (1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^ (1/3)])]/2) - 180180*(9/17 - (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^ (2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/ 17 - Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ (85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/ 3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^3 + 90090* (-9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^2 + 109395* (-9/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^4)^2), 32768/ ((8/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 - Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/2)* (315 - 13860*(9/17 - (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/ 17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ (85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^ (1/3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^ (1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^ (1/3)])]/2) + 90090*(9/17 - (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^ (2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/ 17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ (85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/ 3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^2 - 180180* (9/17 - (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^3 + 109395* (9/17 - (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^4)^2), 32768/ ((8/17 + (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 - Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/2)* (315 - 13860*(9/17 - (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/ 17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ (85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^ (1/3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^ (1/3))/143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^ (1/3)])]/2) + 90090*(9/17 - (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^ (2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/ 17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ (85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/ 3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^2 - 180180* (9/17 - (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^3 + 109395* (9/17 - (2*Sqrt[(12 + (17*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/143^(2/3) + (17*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3))/5])/17 + Sqrt[384/1445 - (16*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/(85*143^(2/3)) - (16*42^(2/3))/(85*(143*(66 + (5*I)*Sqrt[66]))^(1/3)) + 384/(3757*Sqrt[60 + (85*(42*(66 + (5*I)*Sqrt[66]))^(1/3))/ 143^(2/3) + (85*42^(2/3))/(143*(66 + (5*I)*Sqrt[66]))^(1/3)])]/ 2)^4)^2)}

f[9,x]

x^17*a[1] + x^16*a[2] + x^15*a[3] + x^14*a[4] + x^13*a[5] + x^12*a[6] + x^11*a[7] + x^10*a[8] + x^9*a[9] + x^8*a[10] + x^7*a[11] + x^6*a[12] + x^5*a[13] + x^4*a[14] + x^3*a[15] + x^2*a[16] + x*a[17] + a[18]

Integrate[f[9,x],{x,-1,1}]

(2*a[2])/17 + (2*a[4])/15 + (2*a[6])/13 + (2*a[8])/11 + (2*a[10])/9 + (2*a[12])/7 + (2*a[14])/5 + (2*a[16])/3 + 2*a[18]

Timing[FullSimplify[g[9] == Integrate[f[9,x],{x,-1,1}]]]

no result after 6.25 days, manually aborted.

again, omitting "Cubics->True, Quartics->True" mysteriously results in success:

Timing[FullSimplify[g[9] == Integrate[f[9,x],{x,-1,1}]]]

{1164.994788, True}

numeric evaluation of a selected polynomial:

nf[9,x]//Factor//Expand

592470169067385 + 9934306144714350*x + 78404769301414500*x^2 + 386827215552330000*x^3 + 1336214342154562500*x^4 + 3429447712842375000*x^5 + 6771888741932343750*x^6 + 10508444636943750000*x^7 + 12973535419605468750*x^8 + 12815684906445312500*x^9 + 10129468943945312500*x^10 + 6369659214843750000*x^11 + 3147586469726562500*x^12 + 1196633593750000000*x^13 + 337997589111328125*x^14 + 66838378906250000*x^15 + 8261871337890625*x^16 + 480651855468750*x^17

Integrate[nf[9,x],{x,-1,1}]

7777777777777777770

N[gn[9],25] //InputForm

7.77777777777777777`25.15051499783199*^18 + 0``6.259659467257056*I

N[Integrate[nf[9,x],{x,-1,1}],25] //InputForm

7.77777777777777777`25.*^18