## Tuesday, May 11, 2021

### [mljfvegs] incommensurate close numbers

thanks to deep mathematics (that also seems obvious), the square roots of squarefree integers are incommensurate with each other: no nontrivial integer linear combination yields an integer.  (if we include 1 among the squarefree numbers, then no nontrivial integer linear combination yields zero.)

approximate those square roots by fractions and divide, yielding numbers close to 1.  this set is also incommensurate.

(or subtract, yielding numbers close to zero.)

here is a quick and dirty Pari/GP function to find the best fraction that approximates a square root within a certain denominator bound.  (better would have been to use continued fractions.)  the denominator bound 2*x was inspired by the derivative of square root: D(sqrt) = 1/(2*sqrt).  this bound causes consecutive entries to typically not have the same approximation (haven't found any exceptions yet), which seems aesthetically nice.  (Pell equation is another way to mechanically chose a fractional square root approximation.)

bestf(x)=my(best=9999); my(f); for(d=1,2*x, my(n=round(x*d)); my(q=abs(n/d-x)); if(q<best,best=q;f=n/d)); f

for(i=2,100, if(issquarefree(i), y=sqrt(i); l=bestf(y); print("sqrt(",i,")/(",l,") = ",y/l)))

output given below.

as numbers get larger, the ratio gets closer to 1.  gotta pack these numbers in somewhere.

inspired by the following.  one has an arbitrarily large number of devices, all ticking at approximately 1 Hz.  start them together simultaneously but avoid any future ticks from coinciding.  however, a better way would be to define a way to break ties, for example by ordering.

sqrt(2)/(3/2) = 0.94280904158206336586779248280646538571
sqrt(3)/(5/3) = 1.0392304845413263761164678049035234202
sqrt(5)/(9/4) = 0.99380798999990653173741051943612277131
sqrt(6)/(5/2) = 0.97979589711327123927891362988235655679
sqrt(7)/(8/3) = 0.99215674164922147143810590761472265964
sqrt(10)/(19/6) = 0.99861399794790926273649269824191111591
sqrt(11)/(10/3) = 0.99498743710661995473447982100120600518
sqrt(13)/(18/5) = 1.0015420209622192480886725742973599851
sqrt(14)/(15/4) = 0.99777530313971770282233299528441314713
sqrt(15)/(27/7) = 1.0041067934611821554168465851287702695
sqrt(17)/(33/8) = 0.99954075772549346662337208629674594549
sqrt(19)/(35/8) = 0.99631975852358252622559588202505500780
sqrt(21)/(32/7) = 1.0024384332715900014411353236280018570
sqrt(22)/(14/3) = 1.0050890913907349045497778814738142030
sqrt(23)/(24/5) = 0.99913156735681657116613293003389456666
sqrt(26)/(51/10) = 0.99980774776329114314278904098485921364
sqrt(29)/(43/8) = 1.0018911269087449360466438123795961965
sqrt(30)/(11/2) = 0.99585919546393838810358142327418569810
sqrt(31)/(39/7) = 0.99934232153359367832913587416486786265
sqrt(33)/(23/4) = 0.99905437331096150606097590751633553360
sqrt(34)/(35/6) = 0.99959175340205150929271192186495709883
sqrt(35)/(65/11) = 1.0011827325245504072037017108796582697
sqrt(37)/(73/12) = 0.99990616936409090778077001290992883212
sqrt(38)/(37/6) = 0.99963470318415834328381498077636392842
sqrt(39)/(25/4) = 0.99919967974374371293550289935036711377
sqrt(41)/(32/5) = 1.0004881620988826072637840116596583226
sqrt(42)/(13/2) = 0.99703703052428618937937960555199948580
sqrt(43)/(59/9) = 1.0002872325206441673067286437071866890
sqrt(46)/(61/9) = 1.0006716368545477582226394580267823270
sqrt(47)/(48/7) = 0.99978296255848560155314791965820714007
sqrt(51)/(50/7) = 0.99979997999599899971991597359141713903
sqrt(53)/(51/7) = 0.99923076911693388034668857726841626581
sqrt(55)/(89/12) = 0.99993687466458376836558167741133209811
sqrt(57)/(83/11) = 1.0005804673250391165012473840530998511
sqrt(58)/(99/13) = 1.0000510139013212900363469025561434250
sqrt(59)/(100/13) = 0.99854894722291906285005931282507842150
sqrt(61)/(86/11) = 0.99898542366247905041194128015523390174
sqrt(62)/(63/8) = 0.99987401574753155805524246969041369698
sqrt(65)/(129/16) = 0.99996995327733949176640164096790959766
sqrt(66)/(65/8) = 0.99988164980134896744121643917120496597
sqrt(67)/(131/16) = 0.99973774312945953831495618012017459040
sqrt(69)/(108/13) = 0.99987139090680530632958718225738329753
sqrt(70)/(92/11) = 1.0003543795516120682347709003953328678
sqrt(71)/(59/7) = 0.99971268495312729515998270073591783426
sqrt(73)/(94/11) = 0.99983022551588130687859714455996564660
sqrt(74)/(43/5) = 1.0002703798886775315964504110522922828
sqrt(77)/(79/9) = 0.99967948717125441194503157932590868415
sqrt(78)/(53/6) = 0.99982198486730341752045766720293441104
sqrt(79)/(80/9) = 0.99992187194800374563528718848348187943
sqrt(82)/(163/18) = 0.99998118089861042502042053377738156709
sqrt(83)/(82/9) = 0.99992563673534987728671506027070759404
sqrt(85)/(83/9) = 0.99970963994742151554241528356464022187
sqrt(86)/(102/11) = 1.0000961122593406007615742824891364793
sqrt(87)/(28/3) = 0.99936204140237304059512237953434539107
sqrt(89)/(151/16) = 0.99962713982056729126576533802624007597
sqrt(91)/(124/13) = 1.0000975498726043095954903724437052438
sqrt(93)/(135/14) = 1.0000823011400101477084476641782021080
sqrt(94)/(126/13) = 1.0003148912128932886185353568332466200
sqrt(95)/(39/4) = 0.99967121485220142634240135383585644095
sqrt(97)/(128/13) = 1.0002746204949168858023495968275712364