>

>

We first use modern computers to list a bunch of squares,  just to save headache! Traditionally these would be calculated as needed!

>	lis:=[seq(i^2,i=2..2500)];

$\mathrm{lis}:=\left[4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324,361,400,441,484,529,576,625,676,729,784,841,900,961,1024,1089,1156,1225,1296,1369,1444,1521,1600,1681,1764,1849,1936,2025,2116,2209,2304,2401,2500,2601,2704,2809,2916,3025,3136,3249,3364,3481,3600,3721,3844,3969,4096,4225,4356,4489,4624,4761,4900,5041,5184,5329,5476,5625,5776,5929,6084,6241,6400,6561,6724,6889,7056,7225,7396,7569,7744,7921,8100,8281,8464,8649,8836,9025,9216,9409,9604,9801,10000,10201,10404,10609,10816,11025,11236,11449,11664,11881,12100,12321,12544,12769,12996,13225,13456,13689,13924,14161,14400,14641,14884,15129,15376,15625,15876,16129,16384,16641,16900,17161,17424,17689,17956,18225,18496,18769,19044,19321,19600,19881,20164,20449,20736,21025,21316,21609,21904,22201,22500,22801,23104,23409,23716,24025,24336,24649,24964,25281,25600,25921,26244,26569,26896,27225,27556,27889,28224,28561,28900,29241,29584,29929,30276,30625,30976,31329,31684,32041,32400,32761,33124,33489,33856,34225,34596,34969,35344,35721,36100,36481,36864,37249,37636,38025,38416,38809,39204,39601,40000,40401,40804,41209,41616,42025,42436,42849,43264,43681,44100,44521,44944,45369,45796,46225,46656,47089,47524,47961,48400,48841,49284,49729,50176,50625,51076,51529,51984,52441,52900,53361,53824,54289,54756,55225,55696,56169,56644,57121,57600,58081,58564,59049,59536,60025,60516,61009,61504,62001,62500,63001,63504,64009,64516,65025,65536,66049,66564,67081,67600,68121,68644,69169,69696,70225,70756,71289,71824,72361,72900,73441,73984,74529,75076,75625,76176,76729,77284,77841,78400,78961,79524,80089,80656,81225,81796,82369,82944,83521,84100,84681,85264,85849,86436,87025,87616,88209,88804,89401,90000,90601,91204,91809,92416,93025,93636,94249,94864,95481,96100,96721,97344,97969,98596,99225,99856,100489,101124,101761,102400,103041,103684,104329,104976,105625,106276,106929,107584,108241,108900,109561,110224,110889,111556,112225,112896,113569,114244,114921,115600,116281,116964,117649,118336,119025,119716,120409,121104,121801,122500,123201,123904,124609,125316,126025,126736,127449,128164,128881,129600,130321,131044,131769,132496,133225,133956,134689,135424,136161,136900,137641,138384,139129,139876,140625,141376,142129,142884,143641,144400,145161,145924,146689,147456,148225,148996,149769,150544,151321,152100,152881,153664,154449,155236,156025,156816,157609,158404,159201,160000,160801,161604,162409,163216,164025,164836,165649,166464,167281,168100,168921,169744,170569,171396,172225,173056,173889,174724,175561,176400,177241,178084,178929,179776,180625,181476,182329,183184,184041,184900,185761,186624,187489,188356,189225,190096,190969,191844,192721,193600,194481,195364,196249,197136,198025,198916,199809,200704,201601,202500,203401,204304,205209,206116,207025,207936,208849,209764,210681,211600,212521,213444,214369,215296,216225,217156,218089,219024,219961,220900,221841,222784,223729,224676,225625,226576,227529,228484,229441,230400,231361,232324,233289,234256,235225,236196,237169,238144,239121,240100,241081,242064,243049,244036,245025,246016,247009,248004,249001,250000,251001,252004,253009,254016,255025,256036,257049,258064,259081,260100,261121,262144,263169,264196,265225,266256,267289,268324,269361,270400,271441,272484,273529,274576,275625,276676,277729,278784,279841,280900,281961,283024,284089,285156,286225,287296,288369,289444,290521,291600,292681,293764,294849,295936,297025,298116,299209,300304,301401,302500,303601,304704,305809,306916,308025,309136,310249,311364,312481,313600,314721,315844,316969,318096,319225,320356,321489,322624,323761,324900,326041,327184,328329,329476,330625,331776,332929,334084,335241,336400,337561,338724,339889,341056,342225,343396,344569,345744,346921,348100,349281,350464,351649,352836,354025,355216,356409,357604,358801,360000,361201,362404,363609,364816,366025,367236,368449,369664,370881,372100,373321,374544,375769,376996,378225,379456,380689,381924,383161,384400,385641,386884,388129,389376,390625,391876,393129,394384,395641,396900,398161,399424,400689,401956,403225,404496,405769,407044,408321,409600,410881,412164,413449,414736,416025,417316,418609,419904,421201,422500,423801,425104,426409,427716,429025,430336,431649,432964,434281,435600,436921,438244,439569,440896,442225,443556,444889,446224,447561,448900,450241,451584,452929,454276,455625,456976,458329,459684,461041,462400,463761,465124,466489,467856,469225,470596,471969,473344,474721,476100,477481,478864,480249,481636,483025,484416,485809,487204,488601,490000,491401,492804,494209,495616,497025,498436,499849,501264,502681,504100,505521,506944,508369,509796,511225,512656,514089,515524,516961,518400,519841,521284,522729,524176,525625,527076,528529,529984,531441,532900,534361,535824,537289,538756,540225,541696,543169,544644,546121,547600,549081,550564,552049,553536,555025,556516,558009,559504,561001,562500,564001,565504,567009,568516,570025,571536,573049,574564,576081,577600,579121,580644,582169,583696,585225,586756,588289,589824,591361,592900,594441,595984,597529,599076,600625,602176,603729,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5,4227136,4231249,4235364,4239481,4243600,4247721,4251844,4255969,4260096,4264225,4268356,4272489,4276624,4280761,4284900,4289041,4293184,4297329,4301476,4305625,4309776,4313929,4318084,4322241,4326400,4330561,4334724,4338889,4343056,4347225,4351396,4355569,4359744,4363921,4368100,4372281,4376464,4380649,4384836,4389025,4393216,4397409,4401604,4405801,4410000,4414201,4418404,4422609,4426816,4431025,4435236,4439449,4443664,4447881,4452100,4456321,4460544,4464769,4468996,4473225,4477456,4481689,4485924,4490161,4494400,4498641,4502884,4507129,4511376,4515625,4519876,4524129,4528384,4532641,4536900,4541161,4545424,4549689,4553956,4558225,4562496,4566769,4571044,4575321,4579600,4583881,4588164,4592449,4596736,4601025,4605316,4609609,4613904,4618201,4622500,4626801,4631104,4635409,4639716,4644025,4648336,4652649,4656964,4661281,4665600,4669921,4674244,4678569,4682896,4687225,4691556,4695889,4700224,4704561,4708900,4713241,4717584,4721929,4726276,4730625,4734976,4739329,4743684,4748041,4752400,4756761,4761124,4765489,4769856,4774225,4778596,4782969,4787344,4791721,4796100,4800481,4804864,4809249,4813636,4818025,4822416,4826809,4831204,4835601,4840000,4844401,4848804,4853209,4857616,4862025,4866436,4870849,4875264,4879681,4884100,4888521,4892944,4897369,4901796,4906225,4910656,4915089,4919524,4923961,4928400,4932841,4937284,4941729,4946176,4950625,4955076,4959529,4963984,4968441,4972900,4977361,4981824,4986289,4990756,4995225,4999696,5004169,5008644,5013121,5017600,5022081,5026564,5031049,5035536,5040025,5044516,5049009,5053504,5058001,5062500,5067001,5071504,5076009,5080516,5085025,5089536,5094049,5098564,5103081,5107600,5112121,5116644,5121169,5125696,5130225,5134756,5139289,5143824,5148361,5152900,5157441,5161984,5166529,5171076,5175625,5180176,5184729,5189284,5193841,5198400,5202961,5207524,5212089,5216656,5221225,5225796,5230369,5234944,5239521,5244100,5248681,5253264,5257849,5262436,5267025,5271616,5276209,5280804,5285401,5290000,5294601,5299204,5303809,5308416,5313025,5317636,5322249,5326864,5331481,5336100,5340721,5345344,5349969,5354596,5359225,5363856,5368489,5373124,5377761,5382400,5387041,5391684,5396329,5400976,5405625,5410276,5414929,5419584,5424241,5428900,5433561,5438224,5442889,5447556,5452225,5456896,5461569,5466244,5470921,5475600,5480281,5484964,5489649,5494336,5499025,5503716,5508409,5513104,5517801,5522500,5527201,5531904,5536609,5541316,5546025,5550736,5555449,5560164,5564881,5569600,5574321,5579044,5583769,5588496,5593225,5597956,5602689,5607424,5612161,5616900,5621641,5626384,5631129,5635876,5640625,5645376,5650129,5654884,5659641,5664400,5669161,5673924,5678689,5683456,5688225,5692996,5697769,5702544,5707321,5712100,5716881,5721664,5726449,5731236,5736025,5740816,5745609,5750404,5755201,5760000,5764801,5769604,5774409,5779216,5784025,5788836,5793649,5798464,5803281,5808100,5812921,5817744,5822569,5827396,5832225,5837056,5841889,5846724,5851561,5856400,5861241,5866084,5870929,5875776,5880625,5885476,5890329,5895184,5900041,5904900,5909761,5914624,5919489,5924356,5929225,5934096,5938969,5943844,5948721,5953600,5958481,5963364,5968249,5973136,5978025,5982916,5987809,5992704,5997601,6002500,6007401,6012304,6017209,6022116,6027025,6031936,6036849,6041764,6046681,6051600,6056521,6061444,6066369,6071296,6076225,6081156,6086089,6091024,6095961,6100900,6105841,6110784,6115729,6120676,6125625,6130576,6135529,6140484,6145441,6150400,6155361,6160324,6165289,6170256,6175225,6180196,6185169,6190144,6195121,6200100,6205081,6210064,6215049,6220036,6225025,6230016,6235009,6240004,6245001,6250000\right]$

>	ch:=proc(n,t,s) local ans,tmp; if n<>t^2-s then return(`bad start`); fi; ans:=[n,t+1,s+2*t+1]; if member(ans[3],lis) then print('success') else print('fails',evalf[2](t+1-sqrt(ans[3])));fi; op(ans);end;

>

chh:=proc(n,t,s) local ans,tmp,ct; if n<>t^2-s then return(`bad start`); fi; if type(n,even) then RETURN("Must have odd number",n):fi: ans:=[n,t+1,s+2*t+1]; for ct from 1 to 2000 do; if ans[2]>n-2 then RETURN("It is prime",n);fi: if (member(ans[3],lis)) then return(`success`,ans[1],ans[2]+sqrt(ans[3]),ans[2]-sqrt(ans[3])); fi; #print(member(ans[3],lis)); ans:=[ans[1],ans[2]+1,ans[3]+2*ans[2]+1]; #print(ans,ct); od; print(`it is prime`);ans; end;

>

ch2:=proc(n,t,s,m) local ans,tmp,ct; if n<>t^2-s then return(`bad start`); fi; if type(n,even) then RETURN("Must have odd number",n):fi: ans:=[n,t+1,s+2*t+1]; for ct from 1 to m do; if ans[2]>n-2 then RETURN("It is prime",n);fi: if (member(ans[3],lis)) then return(`success`,ans[1],ans[2]+sqrt(ans[3]),ans[2]-sqrt(ans[3])); fi; if ct>2500 then RETURN("Limit of 2500 exceeded",ct):fi: #print(member(ans[3],lis)); ans:=[ans[1],ans[2]+1,ans[3]+2*ans[2]+1]; #print(ans,ct); od; print(`it is prime`);ans; end;

>	tch:=proc(n) local t,s; t:=ceil(sqrt(n));s:=t^2-n; ch2(n,t,s,n);end:

>

tch(123);

$\mathrm{success},123,41,3$

>

>	tch(4211);ifactor(4211);

Error, (in ch2) invalid input: member received lis, which is not valid for its 2nd argument, s

$\left(4211\right)$

>

tch(127);

$\mathrm{success},127,127,1$

>	ifactor(4211);

$\left(4211\right)$

>

tch(127);

$\mathrm{success},127,127,1$

>

tch(129);

$\mathrm{success},129,43,3$

>

tch(131);

$\mathrm{success},131,131,1$

>	tch(2^17+1);

$\text{Limit of 2500 exceeded},2501$

>

#

>	tch(23111);

$\mathrm{success},23111,191,121$

>