Problems & Puzzles: Puzzles

Puzzle 1173  F(p)=(p^2+1)/2 with p prime Sebastián Martín Ruiz  sent the following nice puzzle: F(p)= (p^2+1)/2 is a square in the following cases: p=7, F(p)=25=5^2 p= 41, F(p)=841=29^2 p=239, F(p)=28561=13^4 p=9369319, F(p)=43892069261881=37^2*179057^2 Q) Find more terms in this series

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From April 27 to May 3, 2024, contributions came from Giorgos Kalogeropoulos, Michael Branicky, Paul Cleary, Emmanuel Vantieghem, Gennady Gusev, Shyam Sunder Gupta, V. F. Izquierdo, J-M Rebert, Ken Wilke, Mohamed Raja, Adam Stinchcombe, Simon Cavegn

***

Giorgos Kalogeropoulos wrote:

This is  A088165

So, we can find more terms and more information about these numbers there.

***

Michael Branicky wrote:

7, 41, 239 and 9369319 are all prime terms in OEIS A086395: Primes found among the numerators of the continued fraction rational approximations to sqrt(2).

The following terms in A086395 are also in the series, with F(p) square:

 

63018038201

489133282872437279

19175002942688032928599

123426017006182806728593424683999798008235734137469123231828679

118119373866262081485223899617886417593271902402645222441921910431040354300019446804767966058309121

70393657629419512300176098237919643553997207020010109405976727793670773944409421814790772590069113145391
735749901577009096149305609025491601824219052328961940201

228974524552809050788283360051156249247289853445354827969699100229734715189068108939562801569352522895929
715782892801204761955061870146621251580672519463200522350228503627749598922100083501854376556678063361693
192383119263372332155965842418829061431628302088077180458132394612442738848832853798158092286918974225934
50549445039659091869534560354914296851424561

1540082618098998669024285822687063977120001250694755122499946279676784700739427027858889641254834107747247
9476188898758454807130149070596399275669352955595558902646895915644155499979601804654422207852166810541196
8897815508782369617320598616176357046631308873299365244174183560089669343957767158708822118475584411679552
202834200395057475221548485001826312210106279

1524292152515359643868604110650015564099014244540971352096171632009871930662567358364858506408255703763257
4007016542862464585413855325419987306924172938043329571057741136881712317512118795220089192610068501585848
2058799673374817150782941955651966975242123399157996320587987175805401632865439263435686054321354676772189
4269896651385556977561402072422869791155369716198039140057596727796266385907831644985674318031764589999985
5793529475331403866733038463348701179514835110098473935045744253645698052307463456458313291050444620470613
419864675747501626402069994160092799834761
226750279789940552156337993282001628523627588887520533905974780579870490518881404209400460416710202911198
349618801676120820476493282824887774890673285300537876417838867189143252632916574265784789365838354100871
761640089019332576813794499158780206732712206309396501658841531166578576696276454264097764026752302409975
074070290058039914263155318198754202212110831743515736199366564029798515324035055621500362780537164798797
665123250850726995521535010942659138033937358039415426614194851847382392600962252892759174330435692573276
422891908666663512497240265395844190788270017519220012144761949970222459895368795986544375247297373061777
41930889373223532263308052774672760255302595944667374444996689881229923030796325067791266157267481

***

Paul Cleary wrote:

I found another 6 terms, the numbers are getting quite large and not easy to read.

 

p=7,= F(p)=25={{5,2}}
p=41,= F(p)=841={{29,2}}
p=239,= F(p)=28561={{13,4}}
p=9369319,= F(p)=43892069261881={{37,2},{179057,2}}
p=63018038201,= F(p)=1985636569351347658201={{44560482149,2}}
p=489133282872437279,= F(p)=119625684206783871996993080894461921={{3761,2},{91962100830401,2}}
p=19175002942688032928599,= F(p)=183840368926047361112315395593676258316051401={{13558774610046711780701,2}}
p=123426017006182806728593424683999798008235734137469123231828679,= F(p)=761699083700526370841019997065714
8760082041163162495367483045967299990423693775194206356748206283254879235715250785203442521={{820541,2},{41
2124309,2},{168867731761219532933,2},{1528327854129402847802890657,2}}
p=118119373866262081485223899617886417593271902402645222441921910431040354300019446804767966058309121
= F(p)=69760932412788987939145526970360288829799554364947343650710964464063996624263485952103800910808
9582340861460517653553481740260456140958304086239590187330894155816455279775980856708117862795896321=
{{1640689,2},{623700659351936833,2},{81621330294770651861320888171562587614620304100785844883920376945100294097,2}}
p=70393657629419512300176098237919643553997207020010109405976727793670773944409421814790772590069113145391
735749901577009096149305609025491601824219052328961940201
= F(p)=2477633517223965956050338531054185093930627287566043737847149124261894055909932938710328964465045117895198
705516619830550111238942567423305140604894483437029734548669349415291101042442001398020871731970273790418605504
345482069492893509277211721834849342937929396452040141273945863173768429043815802351656888502403149960201
={{1081818484868497,2},{189222836424246529,2},{173241429245578164617,2},{1403585359808090345497967122578396075750569180352430137427281089083490283804221301594329934327712251251730
181,2}}

***

Emmanuel Vantieghem wrote:

***

Gennady Gusev wrote:

This equation reduces to the Pell equation: (p^2+1)/2=a^2 -> 2*a^2-p^2=1.

The first solution is p=7, a=5. The following solutions are determined by the recurrent formula: p=3*p+4*a, a=2*p+3*a.

By the condition of the problem, p must be a prime number. After 2000 iterations (it took 10 sec.) , 14 solutions were found:

p=7, a=5

p=41, a=29

p=239, a=169

p=9369319, a=6625109

p=63018038201, a=44560482149

p=489133282872437279, a=345869461223138161

p=19175002942688032928599, a=13558774610046711780701

p=123426017006182806728593424683999798008235734137469123231828679, 

  a=87275373599917999482560755526644279276078854297832438763690789

p=118119373866262081485223899617886417593271902402645222441921\

  910431040354300019446804767966058309121, a=83523010250342981\

  71990535422887155607588204233065687979373480566479378860549397\

  7036442673881477889

p=703936576294195123001760982379196435539972070200101094059767\

  27793670773944409421814790772590069113145391735749901577009096\

  149305609025491601824219052328961940201, a=49775832662286684\

  80545624692373233139290457521141816865114894498417997932288780\

  90010364391137803439609184849716888714034512509460527841092881\

  20303097849764975101

p=228974524552809050788283360051156249247289853445354827969699\

  10022973471518906810893956280156935252289592971578289280120476\

  19550618701466212515806725194632005223502285036277495989221000\

  83501854376556678063361693192383119263372332155965842418829061\

  43162830208807718045813239461244273884883285379815809228691897\

  422593450549445039659091869534560354914296851424561, a=16190\

  94390302569031167554421436134379523476605462630016900437774609\

  30793431158742114200996546909498159847837140668920394000474798\

  53611365699295650412376047361934416970855010379495856551716751\

  59698118394352331256272064182096515019202466059939532037156743\

  29018248548057410132993955374829889147237990445084913232505251\

  84609976983183669368792435242099462994295481

p=154008261809899866902428582268706397712000125069475512249994\

  62796767847007394270278588896412548341077472479476188898758454\

  80713014907059639927566935295559555890264689591564415549997960\

  18046544222078521668105411968897815508782369617320598616176357\

  04663130887329936524417418356008966934395776715870882211847558\

  4411679552202834200395057475221548485001826312210106279, a=

  10890028628453338879561863821306215525438004539101362555955616\

  83947419411028590812788338293519176941758089140435788660549947\

  52588460415889380357156261948996362400328611148078760868240983\

  28017659139289964553631078957822420473574068246761092296401321\

  80224870853817402246509111647060758535441728493381810278856521\

  62443421249932898756000553002144243166691962503308661

p=152429215251535964386860411065001556409901424454097135209617\

  16320098719306625673583648585064082557037632574007016542862464\

  58541385532541998730692417293804332957105774113688171231751211\

  87952200891926100685015858482058799673374817150782941955651966\

  97524212339915799632058798717580540163286543926343568605432135\

  46767721894269896651385556977561402072422869791155369716198039\

  14005759672779626638590783164498567431803176458999998557935294\

  75331403866733038463348701179514835110098473935045744253645698\

  05230746345645831329105044462047061341986467574750162640206999\

  4160092799834761, a=1077837317553049938348609668714756800735\

  64346431063069314371272240159321401717869578363752047180804272\

  76978208983345327539083290106784346280097886690708486579474807\

  79380108177368455583705613856731786553917345140726733128313690\

  87228420813419793797171892943353998771498759037523140825580857\

  99795634706567616602618154384510626818176478596794869912196445\

  70520158400416234016494503704217046632412600430479567796881976\

  16215415077853474111409031986647785213615421865828369075963740\

  93354959576255564880446331251472947296840960792737049721799484\

  181245489549002501104613262518182669

p=226750279789940552156337993282001628523627588887520533905974\

  78057987049051888140420940046041671020291119834961880167612082\

  04764932828248877748906732853005378764178388671891432526329165\

  74265784789365838354100871761640089019332576813794499158780206\

  73271220630939650165884153116657857669627645426409776402675230\

  24099750740702900580399142631553181987542022121108317435157361\

  99366564029798515324035055621500362780537164798797665123250850\

  72699552153501094265913803393735803941542661419485184738239260\

  09622528927591743304356925732764228919086666635124972402653958\

  44190788270017519220012144761949970222459895368795986544375247\

  29737306177741930889373223532263308052774672760255302595944667\

  374444996689881229923030796325067791266157267481, a=16033666\

  04754139233686429885490882739765641688853937240022966202572824\

  05469162045889199030324126111866433122445032047736527160736259\

  96818601415464556837159848695320696330094678715653042523918140\

  33051284305894511323167358092021516179716154679185568475052097\

  47100331584582636277962485155169642042517659018902582889265257\

  27445507451457668170228115341121827358266956215271426565250868\

  99128901269941340545472525972165925109736470954936768641055870\

  21115213661494251998398142883488007636573915432207710238140603\

  57362494791980747192522753406501225311320550753910225186521286\

  74370668517001483361552427480650018911337824579714606971874425\

  31668132853196463910429936247550023411222977557713669646035276\

  03617398356705321445318830929291530709

***

Shyam Sunder Gupta wrote:

Puzzle 1173 F(p)=(p^2+1)/2 with p prime:

The primes p, such that (p^2+1)/2 is a square are called NSW primes.
The first fourteen such primes including the four above as given in
OEIS A088165 are:
7, 41, 239, 9369319, 63018038201, 489133282872437279,
19175002942688032928599,
123426017006182806728593424683999798008235734137469123231828679,
118119373866262081485223899617886417593271902402645222441921910431040
354300019446804767966058309121,
703936576294195123001760982379196435539972070200101094059767277936707
739444094218147907725900691131453917357499015770090961493056090254916
01824219052328961940201,
228974524552809050788283360051156249247289853445354827969699100229734
715189068108939562801569352522895929715782892801204761955061870146621
251580672519463200522350228503627749598922100083501854376556678063361
693192383119263372332155965842418829061431628302088077180458132394612
442738848832853798158092286918974225934505494450396590918695345603549
14296851424561,
154008261809899866902428582268706397712000125069475512249994627967678
470073942702785888964125483410774724794761888987584548071301490705963
992756693529555955589026468959156441554999796018046544222078521668105
411968897815508782369617320598616176357046631308873299365244174183560
089669343957767158708822118475584411679552202834200395057475221548485
001826312210106279,
152429215251535964386860411065001556409901424454097135209617163200987
193066256735836485850640825570376325740070165428624645854138553254199
873069241729380433295710577411368817123175121187952200891926100685015
858482058799673374817150782941955651966975242123399157996320587987175
805401632865439263435686054321354676772189426989665138555697756140207
242286979115536971619803914005759672779626638590783164498567431803176
458999998557935294753314038667330384633487011795148351100984739350457
442536456980523074634564583132910504446204706134198646757475016264020
69994160092799834761,
226750279789940552156337993282001628523627588887520533905974780579870
490518881404209400460416710202911198349618801676120820476493282824887
774890673285300537876417838867189143252632916574265784789365838354100
871761640089019332576813794499158780206732712206309396501658841531166
578576696276454264097764026752302409975074070290058039914263155318198
754202212110831743515736199366564029798515324035055621500362780537164
798797665123250850726995521535010942659138033937358039415426614194851
847382392600962252892759174330435692573276422891908666663512497240265
395844190788270017519220012144761949970222459895368795986544375247297
373061777419308893732235322633080527746727602553025959446673744449966
89881229923030796325067791266157267481.

***

V. F. Izquierdo wrote:

This sequence continues with:

p=63018038201

p=489133282872437279

p=19175002942688032928599

p=123426017006182806728593424683999798008235734137469123231828679

p=703936576294195123001760982379196435539972070200101094059767277936707739444094218147907725900691131453917
35749901577009096149305609025491601824219052328961940201

p=2289745245528090507882833600511562492472898534453548279696991002297347151890681089395628015693525228959297
157828928012047619550618701466212515806725194632005223502285036277495989221 000835018543765566780633616931923
8311926337233215596584241882906143162830208807718045813239461244273884883285379815809228691897422593450549445
039659091869534560354914296851424561

p=15400826180989986690242858226870639771200012506947551224999462796767847007394270278588896412548341077472479
4761888987584548071301490705963992756693529555955589026468959156441554999796018046544222078521668105411968897
8155087823696173205986161763570466313088732993652441741835600896693439577671587088221184755844116795522028342
00395057475221548485001826312210106279

p=15242921525153596438686041106500155640990142445409713520961716320098719306625673583648585064082557037632574
0070165428624645854138553254199873069241729380433295710577411368817123175121187952200891926100685015858482058
7996733748171507829419556519669752421233991579963205879871758054016328654392634356860543213546767721894269896
6513855569775614020724228697911553697161980391400575967277962663859078316449856743180317645899999855793529475
3314038667330384633487011795148351100984739350457442536456980523074634564583132910504446204706134198646757475
01626402069994160092799834761

p=22675027978994055215633799328200162852362758888752053390597478057987049051888140420940046041671020291119834
9618801676120820476493282824887774890673285300537876417838867189143252632916574265784789365838354100871761640
0890193325768137944991587802067327122063093965016588415311665785766962764542640977640267523024099750740702900
5803991426315531819875420221211083174351573619936656402979851532403505562150036278053716479879766512325085072
6995521535010942659138033937358039415426614194851847382392600962252892759174330435692573276422891908666663512
4972402653958441907882700175192200121447619499702224598953687959865443752472973730617774193088937322353226330
8052774672760255302595944667374444996689881229923030796325067791266157267481

***

J-M Rebert wrote:

Cf. https://oeis.org/A088165

OEIS A088165  NSW primes: NSW numbers that are also prime. 

***

Ken Wilke wrote:

Comment: Let F(p) = r^2.= (p^2+1)//2. This can be rewritten as the Pell equation
p^2– 2*r^2 = -1. The first two solutions for this equation in positive integers are (p,r) = (1,1) and (7,5).
Additional solutions can be found by the recursion relations
p(n) = 6*p(n-1) – p(n-2) and r(n) = 6*r(n-1) – r(n-2)
Using the recursion relation p(n) = 6*p(n-1) – p(n-2), I wrote a small program in UBASIC to generate
value of p(n) and the corresponding values of F(p). After finding the values stated above, I found the
following additional values where p < 10^60 and p is prime:
p = 63018038201, F(p)= 44560482149^2
p= 489133282872437279 F(p) = 345869461223138161^2 = 3761^2*91962100830401^2
p=19175002942688032928599 F(p) =13558774610046711780701^2.
Primality checking was done with UBASIC’s RHO and ECM programs.

***

Mohamed Raja wrote:

Formula used investigate and find numbers mentioned below:  F(p)=(p^2+1)/2 to check if p is correct p=sqrt(2*(n^2)-1) - (n = result of F(p) equation)

p=1, (1^2+1)/2 = 1 = 1^2 (jk :D)

p=7, (7^2+1)/2 = 25 = 5^2

p=41, (41^2+1)/2 = 841 = 29^2

p=239, (239^2+1)/2 = 28561 = 13^4

p=9369319 = (9369319^2+1)/2 = 43892069261881 = 37^2 * 179057^2

p=63018038201 = (63018038201^2+1)/2 = 1985636569351347658201 = 44560482149^2

more numbers attached as Text File in email.

***

Adam Stinchombre wrote:

(p*2+1)/2 = x^2 leads to the Pell equation p^2-2x^2=-1.  The continued fraction for the square root of 2 leads to a recursive process for integer solutions. 
 I found several primes, including primes with size of 11, 18, 23, 63, 99, 161, 359, 363, 572, and 728 digits.  The last one is     22675027978994055215
6337993282001628523627588887520533905974780579870490518881404209400460416710202911198349618801676120820
4764932828248877748906732853005378764178388671891432526329165742657847893658383541008717616400890193325
7681379449915878020673271220630939650165884153116657857669627645426409776402675230240997507407029005803
9914263155318198754202212110831743515736199366564029798515324035055621500362780537164798797665123250850
7269955215350109426591380339373580394154266141948518473823926009622528927591743304356925732764228919086
6666351249724026539584419078827001751922001214476194997022245989536879598654437524729737306177741930889
373223532263308052774672760255302595944667374444996689881229923030796325067791266157267481 

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Simon wrote:

p=63018038201, F(p) = 1985636569351347658201 = 44560482149^2
Searched up to 24012000000001

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