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Old 2020-10-01, 14:36   #9
R. Gerbicz
 
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"Robert Gerbicz"
Oct 2005
Hungary

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Found 47668 positive integers x for that x^2+1 is 757-smooth, downloadable at:
https://drive.google.com/file/d/1etr...ew?usp=sharing
An old link giving all 200-smooth solutions from Filip Najman: https://web.math.pmf.unizg.hr/~fnajman/rezplus1.html .

Estimating that there could be roughly 50 missed solutions, among them maybe 30 could be found by an extended search [and note that most of these missing solutions are large so interesting]. The search is exhaustive up to a trivial bound of x<2^32.

Improving a little my above arctan formula, using "only" 55 terms.
Code:
c=[38700408465202267483521896, 8074294657163898941499704, 18084751995220504885485716, 10617955537629685604246870, 19381616293392725222395433, -16189431078249810956810540, 2663025114949728099146368, -50821290463616663282060093, 30425287250801654451464701, 5340157628142302862996217, 22257467534574983490163989, -14144052764618268971400705, -34131146882264238757385174, 11517420707788714186671327, 8809498563247969729331508, 9575603319763050834812608, 15013272622091770042897526, 29140822288315176159081433, -26490842270571571240632907, 5379280183188158967246185, 5063193213575554175323319, -21300699540591335158945812, 8765725584890890359073192, -17326267072532326425177716, -22969712961519988945581982, 31015402417393992158722336, 10715218695035024392773646, -6113768192429322401565231, -4959518686523357079068044, -880411196551256194486712, 27222064789916150912426297, 7028400065270324699241782, 6395517824304850284055402, 18382148784793948016890109, -2855737811750830775510733, 1068772694019506440335964, -41696581550717723841749863, -16855984420970193840253782, -17092009580615526091509951, -23678688598446556163790278, -26729067208666826400667006, 5535790741910445875384556, 23253488897611068459648879, 24301719811330032094835131, 13721545967545530609700672, 13638635186955032439915208, -18955462927775883750148916, 15175566500788591331215485, -5657889873218460002259652, -15307860016596651362908660, -12509769441463825443360783, 11660996728649361829160489, 31487859219307743314750041, 10708478645651051077334419, 22413720577850484247874341];
t=[100706129803452075294, 106655703945746057991, 113990292132078182007, 140759366414993038318, 149715090987851395482, 151179894086004836582, 155531320434402458222, 171268442677083970343, 177235659472193946346, 186312414964043780693, 242114657461222775367, 242526457156343868609, 277741650285265886109, 279268215504325418912, 293274837014756552545, 302685178196926874954, 306254909186162917405, 311286554505870488322, 363062694467323053757, 397699150117042862902, 398125775635597684856, 418334276059947230443, 422700922123169074432, 503324067165721943132, 506455457999459591693, 521654927153748407703, 525407238990959323474, 547748886534189022833, 647982671411101494018, 649758297700675498335, 650661171274734088043, 750314893492593005877, 754220218301026231032, 817599728075480257318, 821365850240730409698, 895965022987753171419, 903117827229218160068, 904744940324446807318, 1110592749392907292182, 1195553184514347168724, 1350650129695249176568, 1702259183351533337068, 3022165924225178134193, 4256797797404613635163, 5100058866488107804193, 5710642112294212610443, 6322604305061057220059, 6694462477782585046432, 7172059472210548010478, 9443926883403066025057, 14055524716836234863307, 16322365254295693911102, 63405805856857901256461, 68376738690185154260432, 372635354609714721488943];

\p 10000
430143*Pi/2-sum(i=1,length(t),c[i]*atan(1/t[i]))
\p 28                       
sum(i=1,length(t),log(10)/log(t[i]))
vecmin(t)

that gives:
?    realprecision = 10018 significant digits (10000 digits displayed)
? %31 = 2.762575909555579263 E-10012
?    realprecision = 38 significant digits (28 digits displayed)
? %32 = 2.627625297114408276801574520
? %33 = 100706129803452075294
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