[avsarrtp] largest semiprimes below powers of 2

define a semiprime to be a composite that factors into exactly two prime numbers p*q such that p < q < 2*p .

first, here are the 10 largest semiprimes below select powers of two.  previously on difference of squares with one square itself a power of 2, which are elegant but only possible with even powers of two.

2^31 - 267 = 2147483381 = 46271 * 46411
2^31 - 447 = 2147483201 = 44041 * 48761
2^31 - 1237 = 2147482411 = 44257 * 48523
2^31 - 1261 = 2147482387 = 37489 * 57283
2^31 - 1881 = 2147481767 = 41243 * 52069
2^31 - 2181 = 2147481467 = 36979 * 58073
2^31 - 2271 = 2147481377 = 32993 * 65089
2^31 - 2847 = 2147480801 = 36721 * 58481
2^31 - 3465 = 2147480183 = 37171 * 57773
2^31 - 4485 = 2147479163 = 42863 * 50101

2^32 - 83 = 4294967213 = 57139 * 75167
2^32 - 225 = 4294967071 = 65521 * 65551 = (2^16 - 15) * (2^16 + 15)
2^32 - 507 = 4294966789 = 50411 * 85199
2^32 - 1107 = 4294966189 = 53197 * 80737
2^32 - 1125 = 4294966171 = 52289 * 82139
2^32 - 1169 = 4294966127 = 58369 * 73583
2^32 - 1947 = 4294965349 = 47507 * 90407
2^32 - 2073 = 4294965223 = 50333 * 85331
2^32 - 2603 = 4294964693 = 50159 * 85627
2^32 - 3465 = 4294963831 = 49697 * 86423

2^63 - 499 = 9223372036854775309 = 2298974999 * 4011949691
2^63 - 1881 = 9223372036854773927 = 2707496069 * 3406605883
2^63 - 4809 = 9223372036854770999 = 2807010931 * 3285834029
2^63 - 5491 = 9223372036854770317 = 2366171651 * 3898014767
2^63 - 5947 = 9223372036854769861 = 2295911257 * 4017303373
2^63 - 12117 = 9223372036854763691 = 2291667619 * 4024742489
2^63 - 12915 = 9223372036854762893 = 3022102129 * 3051972317
2^63 - 13249 = 9223372036854762559 = 2738730857 * 3367754087
2^63 - 13531 = 9223372036854762277 = 2521547297 * 3657822341
2^63 - 14311 = 9223372036854761497 = 3007208861 * 3067087277

2^64 - 173 = 18446744073709551443 = 3183958073 * 5793651691
2^64 - 993 = 18446744073709550623 = 3421377637 * 5391612979
2^64 - 2087 = 18446744073709549529 = 3873509161 * 4762282289
2^64 - 4389 = 18446744073709547227 = 3489022939 * 5287080193
2^64 - 4487 = 18446744073709547129 = 3252762193 * 5671101353
2^64 - 5249 = 18446744073709546367 = 3719215471 * 4959848177
2^64 - 6713 = 18446744073709544903 = 3051215147 * 6045704149
2^64 - 6959 = 18446744073709544657 = 4280569103 * 4309413919
2^64 - 8487 = 18446744073709543129 = 3490740163 * 5284479283
2^64 - 9659 = 18446744073709541957 = 3230289899 * 5710553743

2^127 - 5751 = 170141183460469231731687303715884099977 = 11502175072681465387 * 14792087790818572571
2^127 - 5919 = 170141183460469231731687303715884099809 = 9580340788290704387 * 17759408273703519307
2^127 - 9021 = 170141183460469231731687303715884096707 = 10431736506123594239 * 16309957921252484413
2^127 - 16369 = 170141183460469231731687303715884089359 = 10912666737151838909 * 15591164612516645051
2^127 - 19501 = 170141183460469231731687303715884086227 = 11551202543355906989 * 14729304834009001343
2^127 - 23521 = 170141183460469231731687303715884082207 = 9900705818113587227 * 17184752944501361741
2^127 - 26677 = 170141183460469231731687303715884079051 = 12167960973623039383 * 13982719358591868397
2^127 - 31147 = 170141183460469231731687303715884074581 = 11642369228673253363 * 14613965604306664087
2^127 - 41341 = 170141183460469231731687303715884064387 = 12495035573439748261 * 13616702606444136967
2^127 - 43867 = 170141183460469231731687303715884061861 = 10289627824152560981 * 16535212581848830481

2^128 - 1967 = 340282366920938463463374607431768209489 = 16247245438514871883 * 20944003597944230483
2^128 - 5817 = 340282366920938463463374607431768205639 = 15934151135006807009 * 21355537802911212071
2^128 - 15783 = 340282366920938463463374607431768195673 = 15644657512172223677 * 21750707336110361549
2^128 - 18335 = 340282366920938463463374607431768193121 = 16125622675049508359 * 21101967581533637719
2^128 - 20307 = 340282366920938463463374607431768191149 = 15547922442094048129 * 21886034496780526381
2^128 - 21513 = 340282366920938463463374607431768189943 = 15682930096610574179 * 21697626962864609117
2^128 - 25763 = 340282366920938463463374607431768185693 = 17147631346383627497 * 19844278200715037269
2^128 - 29217 = 340282366920938463463374607431768182239 = 13247963057241230723 * 25685636761716575093
2^128 - 43833 = 340282366920938463463374607431768167623 = 17566527694436254297 * 19371065975019875359
2^128 - 71889 = 340282366920938463463374607431768139567 = 14873675459386774777 * 22878162687500775271

the semiprimes above might be useful for testing a factorization algorithm limited to machine-width integers.  this work was inspired by the 'factor' command-line program, which (some versions) have a limit of 2^127-1 .

incidentally, 'factor' (part of GNU coreutils) has supported arbitrary precision since 2008, but the feature was enabled in Debian only in 2020 (Debian Bullseye) because of previous FUD about coreutils depending on GNU MP:

coreutils (8.32-3) unstable; urgency=low

* build with libgmp now that apt pulls it in anyway (Closes: #64527)

-- Michael Stone <mstone@debian.org> Mon, 20 Jul 2020 14:09:06 -0400

Pari/GP source code.  we first try to factor using only Pollard rho (flag 11), quickly rejecting if we find 3 or more factors.  we then try factoring skipping Pollard rho (flag 4) (because we already did it).

allocatemem(10^9);

issemiprime(f)=my(m=matsize(f)); if(m[1]!=2,return(0)); if(f[1,2]!=1,return(0)); if(f[2,2]!=1,return(0)); if(2*f[1,1]<f[2,1],return(0)); 1;

findbefore(e,istart)=my(i); for(i=istart, 2^e, my(n=2^e-i); if((n>6) && (n%2==0),next); my(f=factorint(n,11)); if(issemiprime(f) && ispseudoprime(f[2,1]), my(d=n-2^e); print("dataline "e" "d" "n" "f" POLLARD"); return(-d)); my(m=matsize(f)); if(m[1]==1 && f[1,2]==1, f=factorint(n,4); if(issemiprime(f), my(d=n-2^e); print("dataline "e" "d" "n" "f); return(-d)))); -1;

below is the largest semiprime below each power of two.  not sure what this might be useful for.  (previously, toy RSA.).

2^3 - 2 = 6 = 2 * 3
2^4 - 1 = 15 = 3 * 5 = (2^2 - 1) * (2^2 + 1)
2^5 - 17 = 15 = 3 * 5
2^6 - 29 = 35 = 5 * 7
2^7 - 37 = 91 = 7 * 13
2^8 - 9 = 247 = 13 * 19 = (2^4 - 3) * (2^4 + 3)
2^9 - 19 = 493 = 17 * 29
2^10 - 35 = 989 = 23 * 43
2^11 - 27 = 2021 = 43 * 47
2^12 - 9 = 4087 = 61 * 67 = (2^6 - 3) * (2^6 + 3)
2^13 - 55 = 8137 = 79 * 103
2^14 - 143 = 16241 = 109 * 149
2^15 - 25 = 32743 = 137 * 239
2^16 - 63 = 65473 = 233 * 281
2^17 - 43 = 131029 = 283 * 463
2^18 - 81 = 262063 = 503 * 521 = (2^9 - 9) * (2^9 + 9)
2^19 - 151 = 524137 = 557 * 941
2^20 - 15 = 1048561 = 911 * 1151
2^21 - 51 = 2097101 = 1399 * 1499
2^22 - 215 = 4194089 = 1787 * 2347
2^23 - 45 = 8388563 = 2357 * 3559
2^24 - 9 = 16777207 = 4093 * 4099 = (2^12 - 3) * (2^12 + 3)
2^25 - 403 = 33554029 = 4373 * 7673
2^26 - 65 = 67108799 = 6029 * 11131
2^27 - 279 = 134217449 = 11119 * 12071
2^28 - 209 = 268435247 = 12589 * 21323
2^29 - 51 = 536870861 = 22717 * 23633
2^30 - 137 = 1073741687 = 27779 * 38653
2^31 - 267 = 2147483381 = 46271 * 46411
2^32 - 83 = 4294967213 = 57139 * 75167
2^33 - 171 = 8589934421 = 76493 * 112297
2^34 - 53 = 17179869131 = 125627 * 136753
2^35 - 421 = 34359737947 = 136273 * 252139
2^36 - 3 = 68719476733 = 242819 * 283007
2^37 - 213 = 137438953259 = 297391 * 462149
2^38 - 167 = 274877906777 = 440441 * 624097
2^39 - 187 = 549755813701 = 712321 * 771781
2^40 - 503 = 1099511627273 = 825709 * 1331597
2^41 - 373 = 2199023255179 = 1286533 * 1709263
2^42 - 705 = 4398046510399 = 2014013 * 2183723
2^43 - 769 = 8796093021439 = 2217443 * 3966773
2^44 - 447 = 17592186043969 = 3486391 * 5045959
2^45 - 201 = 35184372088631 = 5591617 * 6292343
2^46 - 225 = 70368744177439 = 8388593 * 8388623 = (2^23 - 15) * (2^23 + 15)
2^47 - 181 = 140737488355147 = 8578639 * 16405573
2^48 - 1893 = 281474976708763 = 15847327 * 17761669
2^49 - 1299 = 562949953420013 = 18463483 * 30489911
2^50 - 365 = 1125899906842259 = 30198439 * 37283381
2^51 - 1081 = 2251799813684167 = 37587401 * 59908367
2^52 - 599 = 4503599627369897 = 48808127 * 92271511
2^53 - 171 = 9007199254740821 = 93220117 * 96622913
2^54 - 177 = 18014398509481807 = 100343501 * 179527307
2^55 - 1261 = 36028797018962707 = 154304077 * 233492191
2^56 - 1529 = 72057594037926407 = 251449199 * 286569193
2^57 - 1111 = 144115188075854761 = 288358229 * 499778309
2^58 - 2157 = 288230376151709587 = 473845213 * 608279599
2^59 - 4005 = 576460752303419483 = 630854771 * 913777273
2^60 - 227 = 1152921504606846749 = 863718871 * 1334834219
2^61 - 2241 = 2305843009213691711 = 1452810757 * 1587159923
2^62 - 827 = 4611686018427387077 = 1792742921 * 2572419037
2^63 - 499 = 9223372036854775309 = 2298974999 * 4011949691
2^64 - 173 = 18446744073709551443 = 3183958073 * 5793651691
2^65 - 1095 = 36893488147419102137 = 5949183623 * 6201437119
2^66 - 291 = 73786976294838206173 = 7036439239 * 10486408507
2^67 - 987 = 147573952589676411941 = 11322007447 * 13034256803
2^68 - 1149 = 295147905179352824707 = 13204545659 * 22351992473
2^69 - 1251 = 590295810358705650461 = 24083972587 * 24509902103
2^70 - 515 = 1180591620717411302909 = 30101795611 * 39219973319
2^71 - 4111 = 2361183241434822602737 = 37656657113 * 62702943449
2^72 - 2043 = 4722366482869645211653 = 57702385861 * 81840055873
2^73 - 1113 = 9444732965739290426279 = 78259949419 * 120684117941
2^74 - 377 = 18889465931478580854407 = 110289293923 * 171271981709
2^75 - 4177 = 37778931862957161705391 = 170344641589 * 221779396819
2^76 - 2283 = 75557863725914323416853 = 195288552673 * 386903700661
2^77 - 2589 = 151115727451828646835683 = 336730384093 * 448773661631
2^78 - 2411 = 302231454903657293674133 = 428192780231 * 705830338243
2^79 - 2167 = 604462909807314587350921 = 736399146137 * 820835973233
2^80 - 159 = 1208925819614629174706017 = 865222140329 * 1397243278073
2^81 - 3105 = 2417851639229258349409247 = 1504976355301 * 1606571180147
2^82 - 1025 = 4835703278458516698823679 = 2178243141013 * 2220001609283
2^83 - 1219 = 9671406556917033397648189 = 2370550533719 * 4079814549131
2^84 - 5319 = 19342813113834066795293497 = 3743220029701 * 5167426162597
2^85 - 5481 = 38685626227668133590592151 = 5445117332089 * 7104645110159
2^86 - 3297 = 77371252455336267181191967 = 8279539254449 * 9344874162383
2^87 - 571 = 154742504910672534362389957 = 12151712952499 * 12734213317543
2^88 - 1523 = 309485009821345068724779533 = 17258876427671 * 17931932656123
2^89 - 3543 = 618970019642690137449558569 = 20667822274579 * 29948487625811
2^90 - 3575 = 1237940039285380274899120649 = 25852344457321 * 47885020305569
2^91 - 14487 = 2475880078570760549798233961 = 44287089457829 * 55905233531509
2^92 - 1655 = 4951760157141521099596495241 = 68900966313353 * 71867789700097
2^93 - 2013 = 9903520314283042199192991779 = 72818547054119 * 136002717919141
2^94 - 1823 = 19807040628566084398385985761 = 127334241902017 * 155551565177633
2^95 - 771 = 39614081257132168796771974397 = 168927433481341 * 234503540607617
2^96 - 3633 = 79228162514264337593543946703 = 226915003627547 * 349153477062749
2^97 - 703 = 158456325028528675187087899969 = 377528015377169 * 419720705681201
2^98 - 7547 = 316912650057057350374175793797 = 504322581218419 * 628392742778663
2^99 - 5005 = 633825300114114700748351597683 = 778215345985117 * 814460037808399
2^100 - 3053 = 1267650600228229401496703202323 = 852649093227773 * 1486720164598351
2^101 - 10533 = 2535301200456458802993406400219 = 1175739633767197 * 2156345782384727
2^102 - 9095 = 5070602400912917605986812812409 = 1606031568230581 * 3157224615764789
2^103 - 1161 = 10141204801825835211973625641847 = 2568128689638119 * 3948869401577713
2^104 - 1889 = 20282409603651670423947251284127 = 3982579711458899 * 5092781833165573
2^105 - 3331 = 40564819207303340847894502568701 = 5099766108348349 * 7954250909840449
2^106 - 425 = 81129638414606681695789005143639 = 6574033780821821 * 12340922045652259
2^107 - 2959 = 162259276829213363391578010285169 = 11959245418571939 * 13567685179972571
2^108 - 5445 = 324518553658426726783156020570811 = 14319028682673241 * 22663447420222771
2^109 - 1863 = 649037107316853453566312041150649 = 22842198789725741 * 28413950569801789
2^110 - 131 = 1298074214633706907132624082304893 = 35367428524858559 * 36702533058668227
2^111 - 1615 = 2596148429267413814265248164608433 = 36566779302877591 * 70997459408822263
2^112 - 2283 = 5192296858534827628530496329217813 = 67641744679950409 * 76761722854760557
2^113 - 23553 = 10384593717069655257060992658416639 = 94092055716457949 * 110366317729980811
2^114 - 411 = 20769187434139310514121985316879973 = 118121996814495551 * 175828279187967323
2^115 - 15865 = 41538374868278621028243970633744903 = 199648814215779179 * 208057208010182357
2^116 - 479 = 83076749736557242056487941267521057 = 247584777165748307 * 335548698460328251
2^117 - 411 = 166153499473114484112975882535042661 = 365157478894991609 * 455018749652654029
2^118 - 3527 = 332306998946228968225951765070082617 = 561114038083576319 * 592227205865651143
2^119 - 747 = 664613997892457936451903530140171541 = 596963533949333113 * 1113324282130885757
2^120 - 527 = 1329227995784915872903807060280344049 = 879069357141345137 * 1512085462866600577
2^121 - 2233 = 2658455991569831745807614120560686919 = 1572613511877682669 * 1690470017897573251
2^122 - 8345 = 5316911983139663491615228241121369959 = 1889325481702291343 * 2814185292387577513
2^123 - 7011 = 10633823966279326983230456482242749597 = 2477236125782494633 * 4292616216760676309
2^124 - 13247 = 21267647932558653966460912964485499969 = 3682723456895302243 * 5774978268525276683
2^125 - 289 = 42535295865117307932921825928971026143 = 5766628885980996871 * 7376111191847811433
2^126 - 797 = 85070591730234615865843651857942052067 = 8526816808278793903 * 9976828826396094989
2^127 - 5751 = 170141183460469231731687303715884099977 = 11502175072681465387 * 14792087790818572571
2^128 - 1967 = 340282366920938463463374607431768209489 = 16247245438514871883 * 20944003597944230483
2^129 - 11619 = 680564733841876926926749214863536411293 = 20286858499088660161 * 33547073533955477213
2^130 - 1715 = 1361129467683753853853498429727072844109 = 28160277140047153109 * 48335087787473199001
2^131 - 1407 = 2722258935367507707706996859454145690241 = 46641953928865273739 * 58365027749894182819
2^132 - 803 = 5444517870735015415413993718908291382493 = 59220030885020465429 * 91937099480847963817
2^133 - 10665 = 10889035741470030830827987437816582755927 = 96246173425464733063 * 113137336830359831729
2^134 - 765 = 21778071482940061661655974875633165532419 = 124801769205687575789 * 174501304120515402671
2^135 - 7339 = 43556142965880123323311949751266331059029 = 167530058442310550731 * 259990018333807268959
2^136 - 9089 = 87112285931760246646623899502532662123647 = 218648816890130244283 * 398411878787039869709
2^137 - 8659 = 174224571863520493293247799005065324256813 = 359664615602929607777 * 484408430257884302669
2^138 - 11085 = 348449143727040986586495598010130648519859 = 527609138899783763051 * 660430455116181832409
2^139 - 2955 = 696898287454081973172991196020261297058933 = 659310130233364802329 * 1057011344884709612477
2^140 - 1077 = 1393796574908163946345982392040522594122699 = 1118232800842855445837 * 1246427911842333182327
2^141 - 1165 = 2787593149816327892691964784081045188246387 = 1201596788839791683051 * 2319907289788868048537
2^142 - 3773 = 5575186299632655785383929568162090376491331 = 2182833199343297599981 * 2554105508982520058351
2^143 - 8601 = 11150372599265311570767859136324180752981607 = 3045575696264389672331 * 3661170731347120600597
2^144 - 13019 = 22300745198530623141535718272648361505967397 = 3979038353983526116379 * 5604556481895864618943
2^145 - 11029 = 44601490397061246283071436545296723011949803 = 6382465847742404555581 * 6988128328620452072263
2^146 - 3521 = 89202980794122492566142873090593446023918143 = 7538355363568267402243 * 11833215136716295201301
2^147 - 9799 = 178405961588244985132285746181186892047833529 = 9759589554582595680193 * 18280068090002291548153
2^148 - 455 = 356811923176489970264571492362373784095686201 = 14910147160155664374409 * 23930811637459706698289
2^149 - 1195 = 713623846352979940529142984724747568191372117 = 21913211640092730527197 * 32565917678964199782361
2^150 - 4683 = 1427247692705959881058285969449495136382741941 = 28020894801156073293331 * 50935121909350130636311
2^151 - 1887 = 2854495385411919762116571938898990272765491361 = 51873925037492254456897 * 55027557358515141846113
2^152 - 225 = 5708990770823839524233143877797980545530986271 = 75557863725914323419121 * 75557863725914323419151 = (2^76 - 15) * (2^76 + 15)
2^153 - 8691 = 11417981541647679048466287755595961091061964301 = 82301380729341854683709 * 138733778710190856783889
2^154 - 857 = 22835963083295358096932575511191922182123945127 = 149555643917543119844741 * 152692084933189623020347
2^155 - 12151 = 45671926166590716193865151022383844364247879817 = 158568344191456332784327 * 288026758426928954253871
2^156 - 863 = 91343852333181432387730302044767688728495783073 = 239875593057055175708333 * 380796775399550572275781
2^157 - 30051 = 182687704666362864775460604089535377456991537821 = 396417831356494304954899 * 460846334891716237979279
2^158 - 3425 = 365375409332725729550921208179070754913983132319 = 516351419879203017920101 * 707609963420266913610419
2^159 - 1185 = 730750818665451459101842416358141509827966270303 = 730835217970956926042851 * 999884516641467022279253
2^160 - 17669 = 1461501637330902918203684832716283019655932525307 = 966117749344982725849439 * 1512757257924083326013413
2^161 - 5049 = 2923003274661805836407369665432566039311865080903 = 1249852263674235439330219 * 2338679026006600554401237
2^162 - 3983 = 5846006549323611672814739330865132078623730167921 = 2369969710902591318905291 * 2466700955050260454362931
2^163 - 315 = 11692013098647223345629478661730264157247460343493 = 2741974592330540054908487 * 4264085134614468524987539
2^164 - 24333 = 23384026197294446691258957323460528314494920663283 = 3817752050526527276191429 * 6125077241218278294154327
2^165 - 5733 = 46768052394588893382517914646921056628989841369499 = 6222261697370625300299677 * 7516246450121492259828887
2^166 - 10353 = 93536104789177786765035829293842113257979682740111 = 8479048115430146786625139 * 11031439321468297693349749
2^167 - 17199 = 187072209578355573530071658587684226515959365483729 = 12620774891258962173453871 * 14822561307857581915065599
2^168 - 10679 = 374144419156711147060143317175368453031918730991177 = 17758672912692800035367773 * 21068264559864486152541149
2^169 - 8323 = 748288838313422294120286634350736906063837461995389 = 25936386688537019303442929 * 28850928515965561647665741
2^170 - 7983 = 1496577676626844588240573268701473812127674923999441 = 29043137000518718910670463 * 51529477569868408972767407
2^171 - 10231 = 2993155353253689176481146537402947624255349848004617 = 54607133527241331388116161 * 54812533819606604666017097
2^172 - 9125 = 5986310706507378352962293074805895248510699696020571 = 66234172813313451556090639 * 90380998995492781352358389
2^173 - 6433 = 11972621413014756705924586149611790497021399392052959 = 103125295357317814614295961 * 116097814522915443190737719
2^174 - 27363 = 23945242826029513411849172299223580994042798784091421 = 114550682381688293445545761 * 209036230323297694477800061
2^175 - 3015 = 47890485652059026823698344598447161988085597568234553 = 185861398998321720351800807 * 257667734721460185172064479
2^176 - 21699 = 95780971304118053647396689196894323976171195136453437 = 245336281373228530483367357 * 390406876504364508725357441
2^177 - 7665 = 191561942608236107294793378393788647952342390272942607 = 363226541215508611047441259 * 527389716531147105807067373
2^178 - 18875 = 383123885216472214589586756787577295904684780545881669 = 525767606210707389647204057 * 728694352201932596407351117
2^179 - 37777 = 766247770432944429179173513575154591809369561091763311 = 668803893876111208193990411 * 1145698727906754173822473901
2^180 - 6827 = 1532495540865888858358347027150309183618739122183595349 = 1218596969303970098175317701 * 1257590146265675684717428049
2^181 - 26893 = 3064991081731777716716694054300618367237478244367177459 = 1748858658739302103515615791 * 1752566490388213879215450749
2^182 - 5025 = 6129982163463555433433388108601236734474956488734403679 = 2277380364545712527892758257 * 2691681310199735796686880847
2^183 - 1665 = 12259964326927110866866776217202473468949912977468815743 = 2525356109279545991370166463 * 4854746735273201771601610561
2^184 - 9819 = 24519928653854221733733552434404946937899825954937624997 = 4866069414947948485688264293 * 5038959900270248657580798529
2^185 - 12643 = 49039857307708443467467104868809893875799651909875256989 = 5053983469398491155404352681 * 9703208885553598661272762069
2^186 - 147 = 98079714615416886934934209737619787751599303819750539117 = 7486239030893178830635938877 * 13101333554896530354388155121
2^187 - 16225 = 196159429230833773869868419475239575503198607639501062303 = 12952028463478026547218224629 * 15145073976942049162118663107
2^188 - 10065 = 392318858461667547739736838950479151006397215279002146991 = 15089149347268625360319448837 * 26000064644643699874318731043
2^189 - 9441 = 784637716923335095479473677900958302012794430558004304671 = 24930846977531896527238364881 * 31472565598371525680493860591
2^190 - 13437 = 1569275433846670190958947355801916604025588861116008614787 = 33824827729144079690745550783 * 46394188505934481491964009789
2^191 - 135 = 3138550867693340381917894711603833208051177722232017256313 = 48554131995870847876553140783 * 64640242522721851491253341911
2^192 - 11343 = 6277101735386680763835789423207666416102355444464034501553 = 77572791882969252814364903173 * 80918858056013701086579696061
2^193 - 7129 = 12554203470773361527671578846415332832204710888928069018663 = 80745776654446990922230094437 * 155478143760995723034651686299
2^194 - 5777 = 25108406941546723055343157692830665664409421777856138045807 = 138614457936767860417911817579 * 181138441943772519746921048333
2^195 - 3759 = 50216813883093446110686315385661331328818843555712276099409 = 171120929373057679871047890403 * 293458047867521030310336889403
2^196 - 11147 = 100433627766186892221372630771322662657637687111424552195189 = 234391324213584263684576506303 * 428486967694541538269125762763
2^197 - 69135 = 200867255532373784442745261542645325315275374222849104343537 = 447304164874116820320039607649 * 449061894133946022052873440913
2^198 - 22811 = 401734511064747568885490523085290650630550748445698208802533 = 608327572274437091441651510071 * 660391751704970520940847817923
2^199 - 30121 = 803469022129495137770981046170581301261101496891396417620567 = 777281759987239587530940907213 * 1033690822929751315709564315059
2^200 - 19727 = 1606938044258990275541962092341162602522202993782792835281649 = 1056464511636976630509885532783 * 1521052554589896058033166035103
2^201 - 8353 = 3213876088517980551083924184682325205044405987565585670594399 = 1291063166015257886428343320279 * 2489325211280947279172077124281
2^202 - 321 = 6427752177035961102167848369364650410088811975131171341205183 = 2389392776391084120195016058107 * 2690119531852095158353338579469
2^203 - 27597 = 12855504354071922204335696738729300820177623950262342682383411 = 3353999001002525317205714079397 * 3832888545950477146532406966263
2^204 - 107 = 25711008708143844408671393477458601640355247900524685364821909 = 4723917437784513359811300435343 * 5442730328538963783774197227163
2^205 - 189 = 51422017416287688817342786954917203280710495801049370729643843 = 6757789778928952039721118279481 * 7609295213151423146895888973403
2^206 - 28841 = 102844034832575377634685573909834406561420991602098741459259223 = 8066624635151595390542877217517 * 12749326947037574463385277419219
2^207 - 4219 = 205688069665150755269371147819668813122841983204197482918571909 = 12873041833206862402305535167701 * 15978202535982193596298358042609
2^208 - 3483 = 411376139330301510538742295639337626245683966408394965837148773 = 14722539268879362332959332619037 * 27941928482396521268431455089129
2^209 - 3859 = 822752278660603021077484591278675252491367932816789931674300653 = 24967480816942018469461465423709 * 32952955273818151657700278046417
2^210 - 1353 = 1645504557321206042154969182557350504982735865633579863348607671 = 32155961170987687621429445220191 * 51172613020998479011346863674281
2^211 - 16245 = 3291009114642412084309938365114701009965471731267159726697201803 = 44035690041943886519769455679797 * 74735041315526882679054126852799
2^212 - 1493 = 6582018229284824168619876730229402019930943462534319453394434603 = 68942582125894371904429267623551 * 95471014086266174584968076666453
2^213 - 7569 = 13164036458569648337239753460458804039861886925068638906788864623 = 108908823148784949202059268127663 * 120872084354319956305424730649921
2^214 - 4851 = 26328072917139296674479506920917608079723773850137277813577739533 = 145977183518461670049100748045909 * 180357452326168472066303200050137
2^215 - 5265 = 52656145834278593348959013841835216159447547700274555627155483503 = 216094311931410950700199356867037 * 243672058573165165899497388735419
2^216 - 11447 = 105312291668557186697918027683670432318895095400549111254310966089 = 247290940328738022362319143845547 * 425863929865604945326560195290587
2^217 - 295 = 210624583337114373395836055367340864637790190801098222508621954777 = 403426174473524207621385295832219 * 522089533759136660813490945347483
2^218 - 26691 = 421249166674228746791672110734681729275580381602196445017243883453 = 637931262197364934929221188924391 * 660336295831041302273685548049083
2^219 - 1879 = 842498333348457493583344221469363458551160763204392890034487818409 = 860933145258310283787156401928773 * 978587405989205239660897687469333
2^220 - 3465 = 1684996666696914987166688442938726917102321526408785780068975637111 = 1121884350899202583210868067359119 * 1501934370816717178660266130134169
2^221 - 16351 = 3369993333393829974333376885877453834204643052817571560137951264801 = 1509918307932190709842801378232549 * 2231904412106230205546338774635149
2^222 - 26933 = 6739986666787659948666753771754907668409286105635143120275902535371 = 1971402331617701126221196069251663 * 3418879321937767527724119141611717
2^223 - 4299 = 13479973333575319897333507543509815336818572211270286240551805120309 = 2630591697010590858988955965836293 * 5124312278828366205617602811014513
2^224 - 8003 = 26959946667150639794667015087019630673637144422540572481103610241213 = 3766730585373261915301648748067937 * 7157386506972348275479420527337949
2^225 - 22059 = 53919893334301279589334030174039261347274288845081144962207220476373 = 5446368538128035827070193655908409 * 9900155113784131704900332561782397
2^226 - 8601 = 107839786668602559178668060348078522694548577690162289924414440988263 = 9922229154826845007133818079081573 * 10868503940582946040714151442701531
2^227 - 2805 = 215679573337205118357336120696157045389097155380324579848828881990923 = 12058556732453844817613271982908631 * 17886018876266928142575099076753133
2^228 - 4545 = 431359146674410236714672241392314090778194310760649159697657763982911 = 17063317704767839842902507173168759 * 25279910632729980264912228152107129
2^229 - 19569 = 862718293348820473429344482784628181556388621521298319395315527955343 = 28542528769577365349343910690007311 * 30225713366657487904106713081597313
2^230 - 3305 = 1725436586697640946858688965569256363112777243042596638790631055946519 = 40881028392685291819680571338840097 * 42206291146197477996094714995868727
2^231 - 22837 = 3450873173395281893717377931138512726225554486085193277581262111876811 = 52938858691434978325441328881267607 * 65186013803384120081558230446734573
2^232 - 8153 = 6901746346790563787434755862277025452451108972170386555162524223791143 = 64552229200095504945296750632089563 * 106917242554039522806028097613930661
2^233 - 1431 = 13803492693581127574869511724554050904902217944340773110325048447597161 = 99314730015445255386316504740368491 * 138987365634830122552497603375957371
2^234 - 10241 = 27606985387162255149739023449108101809804435888681546220650096895186943 = 138773334195614511319319423174924159 * 198935808144866824742907847564181377
2^235 - 11511 = 55213970774324510299478046898216203619608871777363092441300193790382857 = 229220175805067515993861643590583719 * 240877447111284607886951328254564303
2^236 - 5705 = 110427941548649020598956093796432407239217743554726184882600387580783031 = 289718436755859588921903799834681277 * 381156072720717531959848317190249603
2^237 - 21931 = 220855883097298041197912187592864814478435487109452369765200775161555541 = 397221032607618888448923803107865959 * 556002489715752076345224143770038499
2^238 - 13703 = 441711766194596082395824375185729628956870974218904739530401550323141241 = 663747390082079512490221570052664589 * 665481737171085020519140503367741469
2^239 - 4987 = 883423532389192164791648750371459257913741948437809479060803100646304901 = 746955363917410832233879878115166941 * 1182699228178874567616833667659311561
2^240 - 26123 = 1766847064778384329583297500742918515827483896875618958121606201292593653 = 1301478298437490895938315276503066473 * 1357569363161566980347247757596735661
2^241 - 26311 = 3533694129556768659166595001485837031654967793751237916243212402585213241 = 1829620360777656461203642176044500139 * 1931381069707170120943785373650645419
2^242 - 11157 = 7067388259113537318333190002971674063309935587502475832486424805170467947 = 2142083064418158593144018490933247559 * 3299306351144328927940934580264441533
2^243 - 2851 = 14134776518227074636666380005943348126619871175004951664972849610340955357 = 3492441545175241022407276582112795099 * 4047247845208481731749757488022123943
2^244 - 27137 = 28269553036454149273332760011886696253239742350009903329945699220681889279 = 4799431190944108120815778488123925889 * 5890188214344036620522351051481840511
2^245 - 23833 = 56539106072908298546665520023773392506479484700019806659891398441363808999 = 7038948288247863513125159826528922739 * 8032322977468843532284541833436145341
2^246 - 4043 = 113078212145816597093331040047546785012958969400039613319782796882727661621 = 8783979147067066377655672107094843097 * 12873233218406823703341360484704008893
2^247 - 2341 = 226156424291633194186662080095093570025917938800079226639565593765455328987 = 11028138128514059860136503600551344843 * 20507217234329807168642456745839157809
2^248 - 6707 = 452312848583266388373324160190187140051835877600158453279131187530910655949 = 20764798496912634752982493234527345347 * 21782674589908371173902727081232389167
2^249 - 405 = 904625697166532776746648320380374280103671755200316906558262375061821324907 = 27633861869621074350434277400808433547 * 32736130094108244744831471747559710881
2^250 - 27587 = 1809251394333065553493296640760748560207343510400633813116524750123642623037 = 36996636617980930928678594986278944177 * 48903131736406052639523163715879151181
2^251 - 30009 = 3618502788666131106986593281521497120414687020801267626233049500247285271239 = 45674044301681987460809207635394315023 * 79224488305995633302542532294308846793
2^252 - 4307 = 7237005577332262213973186563042994240829374041602535252466099000494570598189 = 68062171619312227903458045901010649751 * 106329336915820738029923241928765982939
2^253 - 1533 = 14474011154664524427946373126085988481658748083205070504932198000989141203459 = 107448486798726318613312015668381380461 * 134706514590358080391744759308799932719
2^254 - 5397 = 28948022309329048855892746252171976963317496166410141009864396001978282404587 = 161518270030011452126577682608336231281 * 179224445036157600027400301248103360027
2^255 - 591 = 57896044618658097711785492504343953926634992332820282019728792003956564819377 = 189534297503607601955633970592627848347 * 305464738473289260609373534577361923491
2^256 - 40919 = 115792089237316195423570985008687907853269984665640564039457584007913129599017 = 284500434698769099480653291815790748659 * 407001449259357401322238439915554856563
2^257 - 22111 = 231584178474632390847141970017375815706539969331281128078915168015826259257761 = 342574526270136478202231377911312046171 * 676011088728813233802107667402361882291
2^258 - 3243 = 463168356949264781694283940034751631413079938662562256157830336031652518556501 = 590751613139587867686008751564080404699 * 784032318570788874254110585935295398799
2^259 - 9439 = 926336713898529563388567880069503262826159877325124512315660672063305037110049 = 846019150495718589846102728112035968993 * 1094935869188953345018612318009054300993
2^260 - 32207 = 1852673427797059126777135760139006525652319754650249024631321344126610074206769 = 1048715229374311851888204876416123272069 * 1766612494892829607459585037113770106301
2^261 - 28509 = 3705346855594118253554271520278013051304639509300498049262642688253220148449443 = 1808703307766230006940951197582136256931 * 2048620600008890057826429398045285969153
2^262 - 8103 = 7410693711188236507108543040556026102609279018600996098525285376506440296947801 = 2270985961139241854812828243453603554617 * 3263205426188833111386514744479586104353

the entries that took the longest to find: 256 = 33 hours, 260 = 34 hours, 261 = 32 hours.