Proving existence of integer solution in $x^2y^3+z^2=6$Perfect square modulo $n = pq$Factor the modulusSquare roots in finite fields, i.e. mod $p^m$Determine the number of solutions of $x^pequiv 1mod p^h$ using primitive rootsExplain solution of this system of non-linear congruence equationsExtended Euclid’s Algorithm of GCD(19,7)Find the smallest integer $a > 2$ such that $2|a, 3|a + 1, 4|a + 2, 5|a + 3, 6|a + 4$What modulus should be chosen when proving impossibility of $|2^m-3^n|=N$?Existence of solutions to DLP and Primitive roots mod $p$$n^2 equiv (p-1) mod p$ where $p$ is a Pythagorean prime.
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Proving existence of integer solution in $x^2y^3+z^2=6$
Perfect square modulo $n = pq$Factor the modulusSquare roots in finite fields, i.e. mod $p^m$Determine the number of solutions of $x^pequiv 1mod p^h$ using primitive rootsExplain solution of this system of non-linear congruence equationsExtended Euclid’s Algorithm of GCD(19,7)Find the smallest integer $a > 2$ such that $2|a, 3|a + 1, 4|a + 2, 5|a + 3, 6|a + 4$What modulus should be chosen when proving impossibility of $|2^m-3^n|=N$?Existence of solutions to DLP and Primitive roots mod $p$$n^2 equiv (p-1) mod p$ where $p$ is a Pythagorean prime.
$begingroup$
I've been working with this curve lately, but I haven't been able to solve this case yet.
$$x^2y^3+z^2=6$$
I achived an algorithm to check for solutions depending on the values of $z$ and got that no solution exist when $|z|leq 10^5$. I could try augmenting the size of the search, but since the algorithm runs in $O(n^2)$ the amount of time it would take would jump to 1 hour just to check $|z|leq 10^6$ and $5$ days for $|z|leq 10^7$.
Right now I know that if a solution exist, then $x,z equiv 1 mod2$ and that $y equiv 1 mod4$ and that $2 not | z,x,y $ and $3 not | z,x,y$, also $(xy,z)=1$.
The question is not to find the values of $x,y$ or $z$, but to prove whether a solution over integers exist or not. Any help, hints or solutions would be appreciated.
EDIT: I think that with python and multiprocessing I've check that for $|z|leq 10^8$ there doesn't exist a solution. But I feel that I've coded something wrong since the program run much faster than it should have.
real-analysis number-theory
asked Mar 22 at 15:10
Erik T.Erik T.
19214
$endgroup$
add a comment |
$begingroup$
I've been working with this curve lately, but I haven't been able to solve this case yet.
$$x^2y^3+z^2=6$$
I achived an algorithm to check for solutions depending on the values of $z$ and got that no solution exist when $|z|leq 10^5$. I could try augmenting the size of the search, but since the algorithm runs in $O(n^2)$ the amount of time it would take would jump to 1 hour just to check $|z|leq 10^6$ and $5$ days for $|z|leq 10^7$.
Right now I know that if a solution exist, then $x,z equiv 1 mod2$ and that $y equiv 1 mod4$ and that $2 not | z,x,y $ and $3 not | z,x,y$, also $(xy,z)=1$.
The question is not to find the values of $x,y$ or $z$, but to prove whether a solution over integers exist or not. Any help, hints or solutions would be appreciated.
EDIT: I think that with python and multiprocessing I've check that for $|z|leq 10^8$ there doesn't exist a solution. But I feel that I've coded something wrong since the program run much faster than it should have.
real-analysis number-theory
asked Mar 22 at 15:10
Erik T.Erik T.
19214
$endgroup$
1
$begingroup$
You can massively improve time by using the fact that $n=x^2y^3$ iff $n$ is powerful - every prime factor of $n$ appears with exponent at least $2$. Using Sage's built-infactorfunction I was able to check $|z|leq 10^6$ in under 20 seconds.
$endgroup$
– Wojowu
Mar 22 at 16:08
$begingroup$
@Wojowu that was the idea I used, but I implemented the factorization algorithm myself. I'll try that implementation instead. Thanks
$endgroup$
– Erik T.
Mar 22 at 16:37
2
$begingroup$
FWIW, from Mathematica, the smallest value of $|y|$ for which this has a solution seems to be $y=-19$ with $(x,z)=(755031379,62531004125)$, but I am not sure about this.
$endgroup$
– Poon Levi
Mar 22 at 17:19
$begingroup$
@PoonLevi Wow, nice solution, how did you find it?
$endgroup$
– Erik T.
Mar 22 at 17:44
$begingroup$
@ErikT. I just did some manual brute forcing with theFindInstancefunction
$endgroup$
– Poon Levi
Mar 23 at 5:36
add a comment |
$begingroup$
I've been working with this curve lately, but I haven't been able to solve this case yet.
$$x^2y^3+z^2=6$$
I achived an algorithm to check for solutions depending on the values of $z$ and got that no solution exist when $|z|leq 10^5$. I could try augmenting the size of the search, but since the algorithm runs in $O(n^2)$ the amount of time it would take would jump to 1 hour just to check $|z|leq 10^6$ and $5$ days for $|z|leq 10^7$.
Right now I know that if a solution exist, then $x,z equiv 1 mod2$ and that $y equiv 1 mod4$ and that $2 not | z,x,y $ and $3 not | z,x,y$, also $(xy,z)=1$.
The question is not to find the values of $x,y$ or $z$, but to prove whether a solution over integers exist or not. Any help, hints or solutions would be appreciated.
EDIT: I think that with python and multiprocessing I've check that for $|z|leq 10^8$ there doesn't exist a solution. But I feel that I've coded something wrong since the program run much faster than it should have.
real-analysis number-theory
asked Mar 22 at 15:10
Erik T.Erik T.
19214
$endgroup$
I've been working with this curve lately, but I haven't been able to solve this case yet.
$$x^2y^3+z^2=6$$
I achived an algorithm to check for solutions depending on the values of $z$ and got that no solution exist when $|z|leq 10^5$. I could try augmenting the size of the search, but since the algorithm runs in $O(n^2)$ the amount of time it would take would jump to 1 hour just to check $|z|leq 10^6$ and $5$ days for $|z|leq 10^7$.
Right now I know that if a solution exist, then $x,z equiv 1 mod2$ and that $y equiv 1 mod4$ and that $2 not | z,x,y $ and $3 not | z,x,y$, also $(xy,z)=1$.
The question is not to find the values of $x,y$ or $z$, but to prove whether a solution over integers exist or not. Any help, hints or solutions would be appreciated.
EDIT: I think that with python and multiprocessing I've check that for $|z|leq 10^8$ there doesn't exist a solution. But I feel that I've coded something wrong since the program run much faster than it should have.
real-analysis number-theory
real-analysis number-theory
asked Mar 22 at 15:10
Erik T.Erik T.
19214
asked Mar 22 at 15:10
Erik T.Erik T.
19214
edited Mar 22 at 15:51
Erik T.
asked Mar 22 at 15:10
Erik T.Erik T.
19214
asked Mar 22 at 15:10
Erik T.Erik T.
19214
asked Mar 22 at 15:10
Erik T.Erik T.
19214
19214
1
$begingroup$
You can massively improve time by using the fact that $n=x^2y^3$ iff $n$ is powerful - every prime factor of $n$ appears with exponent at least $2$. Using Sage's built-infactorfunction I was able to check $|z|leq 10^6$ in under 20 seconds.
$endgroup$
– Wojowu
Mar 22 at 16:08
$begingroup$
@Wojowu that was the idea I used, but I implemented the factorization algorithm myself. I'll try that implementation instead. Thanks
$endgroup$
– Erik T.
Mar 22 at 16:37
2
$begingroup$
FWIW, from Mathematica, the smallest value of $|y|$ for which this has a solution seems to be $y=-19$ with $(x,z)=(755031379,62531004125)$, but I am not sure about this.
$endgroup$
– Poon Levi
Mar 22 at 17:19
$begingroup$
@PoonLevi Wow, nice solution, how did you find it?
$endgroup$
– Erik T.
Mar 22 at 17:44
$begingroup$
@ErikT. I just did some manual brute forcing with theFindInstancefunction
$endgroup$
– Poon Levi
Mar 23 at 5:36
add a comment |
1
$begingroup$
You can massively improve time by using the fact that $n=x^2y^3$ iff $n$ is powerful - every prime factor of $n$ appears with exponent at least $2$. Using Sage's built-infactorfunction I was able to check $|z|leq 10^6$ in under 20 seconds.
$endgroup$
– Wojowu
Mar 22 at 16:08
$begingroup$
@Wojowu that was the idea I used, but I implemented the factorization algorithm myself. I'll try that implementation instead. Thanks
$endgroup$
– Erik T.
Mar 22 at 16:37
2
$begingroup$
FWIW, from Mathematica, the smallest value of $|y|$ for which this has a solution seems to be $y=-19$ with $(x,z)=(755031379,62531004125)$, but I am not sure about this.
$endgroup$
– Poon Levi
Mar 22 at 17:19
$begingroup$
@PoonLevi Wow, nice solution, how did you find it?
$endgroup$
– Erik T.
Mar 22 at 17:44
$begingroup$
@ErikT. I just did some manual brute forcing with theFindInstancefunction
$endgroup$
– Poon Levi
Mar 23 at 5:36
1
1
$begingroup$
You can massively improve time by using the fact that $n=x^2y^3$ iff $n$ is powerful - every prime factor of $n$ appears with exponent at least $2$. Using Sage's built-in
factor function I was able to check $|z|leq 10^6$ in under 20 seconds.$endgroup$
– Wojowu
Mar 22 at 16:08
$begingroup$
You can massively improve time by using the fact that $n=x^2y^3$ iff $n$ is powerful - every prime factor of $n$ appears with exponent at least $2$. Using Sage's built-in
factor function I was able to check $|z|leq 10^6$ in under 20 seconds.$endgroup$
– Wojowu
Mar 22 at 16:08
$begingroup$
@Wojowu that was the idea I used, but I implemented the factorization algorithm myself. I'll try that implementation instead. Thanks
$endgroup$
– Erik T.
Mar 22 at 16:37
$begingroup$
@Wojowu that was the idea I used, but I implemented the factorization algorithm myself. I'll try that implementation instead. Thanks
$endgroup$
– Erik T.
Mar 22 at 16:37
2
2
$begingroup$
FWIW, from Mathematica, the smallest value of $|y|$ for which this has a solution seems to be $y=-19$ with $(x,z)=(755031379,62531004125)$, but I am not sure about this.
$endgroup$
– Poon Levi
Mar 22 at 17:19
$begingroup$
FWIW, from Mathematica, the smallest value of $|y|$ for which this has a solution seems to be $y=-19$ with $(x,z)=(755031379,62531004125)$, but I am not sure about this.
$endgroup$
– Poon Levi
Mar 22 at 17:19
$begingroup$
@PoonLevi Wow, nice solution, how did you find it?
$endgroup$
– Erik T.
Mar 22 at 17:44
$begingroup$
@PoonLevi Wow, nice solution, how did you find it?
$endgroup$
– Erik T.
Mar 22 at 17:44
$begingroup$
@ErikT. I just did some manual brute forcing with the
FindInstance function$endgroup$
– Poon Levi
Mar 23 at 5:36
$begingroup$
@ErikT. I just did some manual brute forcing with the
FindInstance function$endgroup$
– Poon Levi
Mar 23 at 5:36
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
ADDED: works pretty well for $y = -p,$ where $p$ is a prime and $p equiv -5 pmod24.$ Not always, though, there is no solution in integers to $u^2 - 499 v^2 = 6,$ so there cannot be a solution to $z^2 - 499^3 x^2 = 6,$
As in comment by Poon,
$$ 62531004125^2 - 6859 cdot 755031379^2 = 6 $$
$$ 30062417279159968032434332377499679191^2 - 6859 cdot 362989027474817934968547501096752545^2 = 6 $$
are the two smallest solutions with $y = -19$
This is a Pell type situation..
Next is $y = -43.$
$$ 8524568252330795722226762076275477^2 - 79507 cdot 30232197004259457995489433581233^2 = 6 $$
$$ 7854330327479105748865242264224583147937033544548000861492635087111452983686976831425073648904157700425958139623153797892698716197^2 - 79507 cdot 1091798726642836794174438412786249442561116671851858490296127998493079755047027851686091414930149044443059275247490273988265975073^2 = 6 $$
$19 pmod 24,$ or $-5 pmod 24$ looks good, jump to $y=-67$
$$ 282978368450687434487209373191993925035994947255483507^2 - 300763 cdot 515989702066780538602608723807499847473549946045669^2 = 6 $$
$$ 110282895315930497927912550392745703024272661444749938273795258966953853545781325097922911621692636895662478761651920330433345436002171835031498514609892669112804144738962046702905275306924383919904704053765669273732362771438086818759096600035555210393331272621715089906882939770617^2 - 300763 cdot 201092537951519476578846016654832921687447975787740751416540268301538352543770725685782163606655569263151546549701485320486846032648563723182340355499316131383039221816513366591020013230115379554131047351811803703438819429350858064042234604275684238200002441987024851219462842279^2 = 6 $$
Alright, $y=-91$ does not work, so it is probably primes $p equiv -5 pmod 24, $
Looks good: $y = -139$ works.
$$4362811041355070169863445353127930955459611243393190431129982455344444937966635765034690725017372595556332974381641720288818463876300492140964198599085172852063910548429431998295054330240877818521750650943853474767971595481380748602727279462883776818584948606778731999374536671484774081257730189572990126422750941506328066671948299736754296627083409459327142431344285253530904973914932476112673827164600483235886537816825709905101946637717937589365141741638476449995905^2 - 2685619 cdot 2662221602563249187278263196011187615830538137637142943172206847128695186087154957324454563072568300849581804322115608951178431618508361010826207991363516489555641531605807855706702830569730399355959940637296610198308891993171988598931443596581366199909035947670242973782535283742571317010626567192123563836486926084664553687222659057469486167968416946690621995773225236487161717059270246838151043190279154072191947271777597482296371378305290636442120248659630720699^2 = 6 $$
$$ 11007229320114200682613076077054270674493469083551395090272400450528009167811973532453164230028506749238868355270609196939003181071165527762688822477732870085553734659733958280600416037711133666637554796387480108123093268249334915679222608686278074350575513887596916117811796252204388381986306317919874745606321543530836091755198128978452469078117805119958799386852835463813171493204121621031603301970267358021486072087211213638671800520653184667508559060289724518769975994491764389949401683257784292501138366997875642534662126840960900976803282849392676436506058878390727089164075298305914748170136999780085836991430800327841415662720459718357825397573172976708908467441^2 - 2685619 cdot 6716697881848674838156030386241616967270051136862684282955991537338414961773579582932757141004204932054213610185226058428383136198936104632942497489498739897183325583340951152435956413801114255489130165449836905051326165370830554673259419604890586134665267687698856933901633921567579748923093505649829262337020641217111875006923069411778584596026715953822127592173142902673636449348881975583202914018304502788869342977433804925999933660151442708630639152564344393622683488924533630738934557423091516094843646237584273637443239810351573953018195234525958355653029554536372430979361344097937950355948103639632777321526741953229033254611279820660546487889155631849994795^2 = 6 $$
$y = -163 $ works. The others should be checked for class number! No, $139$ has class number 3.
$$
1085997992037838057157328137528530814161327411705546203392088990536040293^2 - 4330747 cdot 521852316521211748718181001351655538497525757942306914225856323090563^2 = 6 $$
$$ 50576439676678868925447962668907434500838480927375612223861414693409656592498369002412775393461481995344704306058443464631047373837930992505267663441324334534575755546957820383354011427592442618572553822492422632438018310472471467898207758462496052055942015486413777383860175293508915160725621756472206275639559775926508637417532793267711813496401310961017300099032326961359121416479939614082943062346592497129301169840684667781108633285175838272455967344224725866133380586503707411632734800342621002287656549985786259734994462255794013879381219457810712405084283774220828444140059155951479917295341262660858326466541291848167815166877848998169467570609203031564636509860322954392994510333265978746217370992701911980229349147987494073342207262560127339452333844537013716461592403989956476819621648310312838004247585673708244092703910695114947918019837457181216465905991224195278477515712580521442345725333566315384496982715121821384411921111090497302852848529853789432402185245489658332331342402733989663728321530595214247562396845918152405928944606618632203463997552998841227239534262770964180949700862983087031814190544748296635415256018314980663318640218710377201213304383260086867630769339280626456705440420391795961162063402561211243834067151983^2 - 4330747 cdot 24303389509168264155926836231677339583236279792966852959122818227864577731252374449598975246414912863111465364548050391636522567455569326936167362733978208903502309971646876492298053014761680209263210334367950807323938188804032444180691208424432641560933430829642849222471126855975046976264861142200196090873689848421843657191122679730405457613374464333528713595028025499158915631933516131901486678847790154871195366492722123901490942121147791994088903595812840508108723450617408520351474791501477137500457342580798802860226722483071140092607989410381522583658260015263828479974612715845634569484085007653986257235669726683443743485813491066481117864482736645186942197877484560957415923165525303520724765971741826705012467970971652276787231995434582749181129198639619283181022181454009410979226141089115582679800903885255370692759102173591511270474888363142061555783426807681868230735428982849314717966630981691678606125268941949114427742580036471039151646099484825259176902899828540485246941295441813001097452114340209278442506644192595084839750875848820948944425473440101086124918167955646103531186497403430181161785931987860726637908855991154397153654665336597089114095843532096882086827889787605551124993774754527399260647535995679182021376817^2 = 6 $$
answered Mar 22 at 17:49
Will JagyWill Jagy
104k5102201
$endgroup$
add a comment |
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$begingroup$
ADDED: works pretty well for $y = -p,$ where $p$ is a prime and $p equiv -5 pmod24.$ Not always, though, there is no solution in integers to $u^2 - 499 v^2 = 6,$ so there cannot be a solution to $z^2 - 499^3 x^2 = 6,$
As in comment by Poon,
$$ 62531004125^2 - 6859 cdot 755031379^2 = 6 $$
$$ 30062417279159968032434332377499679191^2 - 6859 cdot 362989027474817934968547501096752545^2 = 6 $$
are the two smallest solutions with $y = -19$
This is a Pell type situation..
Next is $y = -43.$
$$ 8524568252330795722226762076275477^2 - 79507 cdot 30232197004259457995489433581233^2 = 6 $$
$$ 7854330327479105748865242264224583147937033544548000861492635087111452983686976831425073648904157700425958139623153797892698716197^2 - 79507 cdot 1091798726642836794174438412786249442561116671851858490296127998493079755047027851686091414930149044443059275247490273988265975073^2 = 6 $$
$19 pmod 24,$ or $-5 pmod 24$ looks good, jump to $y=-67$
$$ 282978368450687434487209373191993925035994947255483507^2 - 300763 cdot 515989702066780538602608723807499847473549946045669^2 = 6 $$
$$ 110282895315930497927912550392745703024272661444749938273795258966953853545781325097922911621692636895662478761651920330433345436002171835031498514609892669112804144738962046702905275306924383919904704053765669273732362771438086818759096600035555210393331272621715089906882939770617^2 - 300763 cdot 201092537951519476578846016654832921687447975787740751416540268301538352543770725685782163606655569263151546549701485320486846032648563723182340355499316131383039221816513366591020013230115379554131047351811803703438819429350858064042234604275684238200002441987024851219462842279^2 = 6 $$
Alright, $y=-91$ does not work, so it is probably primes $p equiv -5 pmod 24, $
Looks good: $y = -139$ works.
$$4362811041355070169863445353127930955459611243393190431129982455344444937966635765034690725017372595556332974381641720288818463876300492140964198599085172852063910548429431998295054330240877818521750650943853474767971595481380748602727279462883776818584948606778731999374536671484774081257730189572990126422750941506328066671948299736754296627083409459327142431344285253530904973914932476112673827164600483235886537816825709905101946637717937589365141741638476449995905^2 - 2685619 cdot 2662221602563249187278263196011187615830538137637142943172206847128695186087154957324454563072568300849581804322115608951178431618508361010826207991363516489555641531605807855706702830569730399355959940637296610198308891993171988598931443596581366199909035947670242973782535283742571317010626567192123563836486926084664553687222659057469486167968416946690621995773225236487161717059270246838151043190279154072191947271777597482296371378305290636442120248659630720699^2 = 6 $$
$$ 11007229320114200682613076077054270674493469083551395090272400450528009167811973532453164230028506749238868355270609196939003181071165527762688822477732870085553734659733958280600416037711133666637554796387480108123093268249334915679222608686278074350575513887596916117811796252204388381986306317919874745606321543530836091755198128978452469078117805119958799386852835463813171493204121621031603301970267358021486072087211213638671800520653184667508559060289724518769975994491764389949401683257784292501138366997875642534662126840960900976803282849392676436506058878390727089164075298305914748170136999780085836991430800327841415662720459718357825397573172976708908467441^2 - 2685619 cdot 6716697881848674838156030386241616967270051136862684282955991537338414961773579582932757141004204932054213610185226058428383136198936104632942497489498739897183325583340951152435956413801114255489130165449836905051326165370830554673259419604890586134665267687698856933901633921567579748923093505649829262337020641217111875006923069411778584596026715953822127592173142902673636449348881975583202914018304502788869342977433804925999933660151442708630639152564344393622683488924533630738934557423091516094843646237584273637443239810351573953018195234525958355653029554536372430979361344097937950355948103639632777321526741953229033254611279820660546487889155631849994795^2 = 6 $$
$y = -163 $ works. The others should be checked for class number! No, $139$ has class number 3.
$$
1085997992037838057157328137528530814161327411705546203392088990536040293^2 - 4330747 cdot 521852316521211748718181001351655538497525757942306914225856323090563^2 = 6 $$
$$ 50576439676678868925447962668907434500838480927375612223861414693409656592498369002412775393461481995344704306058443464631047373837930992505267663441324334534575755546957820383354011427592442618572553822492422632438018310472471467898207758462496052055942015486413777383860175293508915160725621756472206275639559775926508637417532793267711813496401310961017300099032326961359121416479939614082943062346592497129301169840684667781108633285175838272455967344224725866133380586503707411632734800342621002287656549985786259734994462255794013879381219457810712405084283774220828444140059155951479917295341262660858326466541291848167815166877848998169467570609203031564636509860322954392994510333265978746217370992701911980229349147987494073342207262560127339452333844537013716461592403989956476819621648310312838004247585673708244092703910695114947918019837457181216465905991224195278477515712580521442345725333566315384496982715121821384411921111090497302852848529853789432402185245489658332331342402733989663728321530595214247562396845918152405928944606618632203463997552998841227239534262770964180949700862983087031814190544748296635415256018314980663318640218710377201213304383260086867630769339280626456705440420391795961162063402561211243834067151983^2 - 4330747 cdot 24303389509168264155926836231677339583236279792966852959122818227864577731252374449598975246414912863111465364548050391636522567455569326936167362733978208903502309971646876492298053014761680209263210334367950807323938188804032444180691208424432641560933430829642849222471126855975046976264861142200196090873689848421843657191122679730405457613374464333528713595028025499158915631933516131901486678847790154871195366492722123901490942121147791994088903595812840508108723450617408520351474791501477137500457342580798802860226722483071140092607989410381522583658260015263828479974612715845634569484085007653986257235669726683443743485813491066481117864482736645186942197877484560957415923165525303520724765971741826705012467970971652276787231995434582749181129198639619283181022181454009410979226141089115582679800903885255370692759102173591511270474888363142061555783426807681868230735428982849314717966630981691678606125268941949114427742580036471039151646099484825259176902899828540485246941295441813001097452114340209278442506644192595084839750875848820948944425473440101086124918167955646103531186497403430181161785931987860726637908855991154397153654665336597089114095843532096882086827889787605551124993774754527399260647535995679182021376817^2 = 6 $$
answered Mar 22 at 17:49
Will JagyWill Jagy
104k5102201
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add a comment |
$begingroup$
ADDED: works pretty well for $y = -p,$ where $p$ is a prime and $p equiv -5 pmod24.$ Not always, though, there is no solution in integers to $u^2 - 499 v^2 = 6,$ so there cannot be a solution to $z^2 - 499^3 x^2 = 6,$
As in comment by Poon,
$$ 62531004125^2 - 6859 cdot 755031379^2 = 6 $$
$$ 30062417279159968032434332377499679191^2 - 6859 cdot 362989027474817934968547501096752545^2 = 6 $$
are the two smallest solutions with $y = -19$
This is a Pell type situation..
Next is $y = -43.$
$$ 8524568252330795722226762076275477^2 - 79507 cdot 30232197004259457995489433581233^2 = 6 $$
$$ 7854330327479105748865242264224583147937033544548000861492635087111452983686976831425073648904157700425958139623153797892698716197^2 - 79507 cdot 1091798726642836794174438412786249442561116671851858490296127998493079755047027851686091414930149044443059275247490273988265975073^2 = 6 $$
$19 pmod 24,$ or $-5 pmod 24$ looks good, jump to $y=-67$
$$ 282978368450687434487209373191993925035994947255483507^2 - 300763 cdot 515989702066780538602608723807499847473549946045669^2 = 6 $$
$$ 110282895315930497927912550392745703024272661444749938273795258966953853545781325097922911621692636895662478761651920330433345436002171835031498514609892669112804144738962046702905275306924383919904704053765669273732362771438086818759096600035555210393331272621715089906882939770617^2 - 300763 cdot 201092537951519476578846016654832921687447975787740751416540268301538352543770725685782163606655569263151546549701485320486846032648563723182340355499316131383039221816513366591020013230115379554131047351811803703438819429350858064042234604275684238200002441987024851219462842279^2 = 6 $$
Alright, $y=-91$ does not work, so it is probably primes $p equiv -5 pmod 24, $
Looks good: $y = -139$ works.
$$4362811041355070169863445353127930955459611243393190431129982455344444937966635765034690725017372595556332974381641720288818463876300492140964198599085172852063910548429431998295054330240877818521750650943853474767971595481380748602727279462883776818584948606778731999374536671484774081257730189572990126422750941506328066671948299736754296627083409459327142431344285253530904973914932476112673827164600483235886537816825709905101946637717937589365141741638476449995905^2 - 2685619 cdot 2662221602563249187278263196011187615830538137637142943172206847128695186087154957324454563072568300849581804322115608951178431618508361010826207991363516489555641531605807855706702830569730399355959940637296610198308891993171988598931443596581366199909035947670242973782535283742571317010626567192123563836486926084664553687222659057469486167968416946690621995773225236487161717059270246838151043190279154072191947271777597482296371378305290636442120248659630720699^2 = 6 $$
$$ 11007229320114200682613076077054270674493469083551395090272400450528009167811973532453164230028506749238868355270609196939003181071165527762688822477732870085553734659733958280600416037711133666637554796387480108123093268249334915679222608686278074350575513887596916117811796252204388381986306317919874745606321543530836091755198128978452469078117805119958799386852835463813171493204121621031603301970267358021486072087211213638671800520653184667508559060289724518769975994491764389949401683257784292501138366997875642534662126840960900976803282849392676436506058878390727089164075298305914748170136999780085836991430800327841415662720459718357825397573172976708908467441^2 - 2685619 cdot 6716697881848674838156030386241616967270051136862684282955991537338414961773579582932757141004204932054213610185226058428383136198936104632942497489498739897183325583340951152435956413801114255489130165449836905051326165370830554673259419604890586134665267687698856933901633921567579748923093505649829262337020641217111875006923069411778584596026715953822127592173142902673636449348881975583202914018304502788869342977433804925999933660151442708630639152564344393622683488924533630738934557423091516094843646237584273637443239810351573953018195234525958355653029554536372430979361344097937950355948103639632777321526741953229033254611279820660546487889155631849994795^2 = 6 $$
$y = -163 $ works. The others should be checked for class number! No, $139$ has class number 3.
$$
1085997992037838057157328137528530814161327411705546203392088990536040293^2 - 4330747 cdot 521852316521211748718181001351655538497525757942306914225856323090563^2 = 6 $$
$$ 50576439676678868925447962668907434500838480927375612223861414693409656592498369002412775393461481995344704306058443464631047373837930992505267663441324334534575755546957820383354011427592442618572553822492422632438018310472471467898207758462496052055942015486413777383860175293508915160725621756472206275639559775926508637417532793267711813496401310961017300099032326961359121416479939614082943062346592497129301169840684667781108633285175838272455967344224725866133380586503707411632734800342621002287656549985786259734994462255794013879381219457810712405084283774220828444140059155951479917295341262660858326466541291848167815166877848998169467570609203031564636509860322954392994510333265978746217370992701911980229349147987494073342207262560127339452333844537013716461592403989956476819621648310312838004247585673708244092703910695114947918019837457181216465905991224195278477515712580521442345725333566315384496982715121821384411921111090497302852848529853789432402185245489658332331342402733989663728321530595214247562396845918152405928944606618632203463997552998841227239534262770964180949700862983087031814190544748296635415256018314980663318640218710377201213304383260086867630769339280626456705440420391795961162063402561211243834067151983^2 - 4330747 cdot 24303389509168264155926836231677339583236279792966852959122818227864577731252374449598975246414912863111465364548050391636522567455569326936167362733978208903502309971646876492298053014761680209263210334367950807323938188804032444180691208424432641560933430829642849222471126855975046976264861142200196090873689848421843657191122679730405457613374464333528713595028025499158915631933516131901486678847790154871195366492722123901490942121147791994088903595812840508108723450617408520351474791501477137500457342580798802860226722483071140092607989410381522583658260015263828479974612715845634569484085007653986257235669726683443743485813491066481117864482736645186942197877484560957415923165525303520724765971741826705012467970971652276787231995434582749181129198639619283181022181454009410979226141089115582679800903885255370692759102173591511270474888363142061555783426807681868230735428982849314717966630981691678606125268941949114427742580036471039151646099484825259176902899828540485246941295441813001097452114340209278442506644192595084839750875848820948944425473440101086124918167955646103531186497403430181161785931987860726637908855991154397153654665336597089114095843532096882086827889787605551124993774754527399260647535995679182021376817^2 = 6 $$
answered Mar 22 at 17:49
Will JagyWill Jagy
104k5102201
$endgroup$
add a comment |
$begingroup$
ADDED: works pretty well for $y = -p,$ where $p$ is a prime and $p equiv -5 pmod24.$ Not always, though, there is no solution in integers to $u^2 - 499 v^2 = 6,$ so there cannot be a solution to $z^2 - 499^3 x^2 = 6,$
As in comment by Poon,
$$ 62531004125^2 - 6859 cdot 755031379^2 = 6 $$
$$ 30062417279159968032434332377499679191^2 - 6859 cdot 362989027474817934968547501096752545^2 = 6 $$
are the two smallest solutions with $y = -19$
This is a Pell type situation..
Next is $y = -43.$
$$ 8524568252330795722226762076275477^2 - 79507 cdot 30232197004259457995489433581233^2 = 6 $$
$$ 7854330327479105748865242264224583147937033544548000861492635087111452983686976831425073648904157700425958139623153797892698716197^2 - 79507 cdot 1091798726642836794174438412786249442561116671851858490296127998493079755047027851686091414930149044443059275247490273988265975073^2 = 6 $$
$19 pmod 24,$ or $-5 pmod 24$ looks good, jump to $y=-67$
$$ 282978368450687434487209373191993925035994947255483507^2 - 300763 cdot 515989702066780538602608723807499847473549946045669^2 = 6 $$
$$ 110282895315930497927912550392745703024272661444749938273795258966953853545781325097922911621692636895662478761651920330433345436002171835031498514609892669112804144738962046702905275306924383919904704053765669273732362771438086818759096600035555210393331272621715089906882939770617^2 - 300763 cdot 201092537951519476578846016654832921687447975787740751416540268301538352543770725685782163606655569263151546549701485320486846032648563723182340355499316131383039221816513366591020013230115379554131047351811803703438819429350858064042234604275684238200002441987024851219462842279^2 = 6 $$
Alright, $y=-91$ does not work, so it is probably primes $p equiv -5 pmod 24, $
Looks good: $y = -139$ works.
$$4362811041355070169863445353127930955459611243393190431129982455344444937966635765034690725017372595556332974381641720288818463876300492140964198599085172852063910548429431998295054330240877818521750650943853474767971595481380748602727279462883776818584948606778731999374536671484774081257730189572990126422750941506328066671948299736754296627083409459327142431344285253530904973914932476112673827164600483235886537816825709905101946637717937589365141741638476449995905^2 - 2685619 cdot 2662221602563249187278263196011187615830538137637142943172206847128695186087154957324454563072568300849581804322115608951178431618508361010826207991363516489555641531605807855706702830569730399355959940637296610198308891993171988598931443596581366199909035947670242973782535283742571317010626567192123563836486926084664553687222659057469486167968416946690621995773225236487161717059270246838151043190279154072191947271777597482296371378305290636442120248659630720699^2 = 6 $$
$$ 11007229320114200682613076077054270674493469083551395090272400450528009167811973532453164230028506749238868355270609196939003181071165527762688822477732870085553734659733958280600416037711133666637554796387480108123093268249334915679222608686278074350575513887596916117811796252204388381986306317919874745606321543530836091755198128978452469078117805119958799386852835463813171493204121621031603301970267358021486072087211213638671800520653184667508559060289724518769975994491764389949401683257784292501138366997875642534662126840960900976803282849392676436506058878390727089164075298305914748170136999780085836991430800327841415662720459718357825397573172976708908467441^2 - 2685619 cdot 6716697881848674838156030386241616967270051136862684282955991537338414961773579582932757141004204932054213610185226058428383136198936104632942497489498739897183325583340951152435956413801114255489130165449836905051326165370830554673259419604890586134665267687698856933901633921567579748923093505649829262337020641217111875006923069411778584596026715953822127592173142902673636449348881975583202914018304502788869342977433804925999933660151442708630639152564344393622683488924533630738934557423091516094843646237584273637443239810351573953018195234525958355653029554536372430979361344097937950355948103639632777321526741953229033254611279820660546487889155631849994795^2 = 6 $$
$y = -163 $ works. The others should be checked for class number! No, $139$ has class number 3.
$$
1085997992037838057157328137528530814161327411705546203392088990536040293^2 - 4330747 cdot 521852316521211748718181001351655538497525757942306914225856323090563^2 = 6 $$
$$ 50576439676678868925447962668907434500838480927375612223861414693409656592498369002412775393461481995344704306058443464631047373837930992505267663441324334534575755546957820383354011427592442618572553822492422632438018310472471467898207758462496052055942015486413777383860175293508915160725621756472206275639559775926508637417532793267711813496401310961017300099032326961359121416479939614082943062346592497129301169840684667781108633285175838272455967344224725866133380586503707411632734800342621002287656549985786259734994462255794013879381219457810712405084283774220828444140059155951479917295341262660858326466541291848167815166877848998169467570609203031564636509860322954392994510333265978746217370992701911980229349147987494073342207262560127339452333844537013716461592403989956476819621648310312838004247585673708244092703910695114947918019837457181216465905991224195278477515712580521442345725333566315384496982715121821384411921111090497302852848529853789432402185245489658332331342402733989663728321530595214247562396845918152405928944606618632203463997552998841227239534262770964180949700862983087031814190544748296635415256018314980663318640218710377201213304383260086867630769339280626456705440420391795961162063402561211243834067151983^2 - 4330747 cdot 24303389509168264155926836231677339583236279792966852959122818227864577731252374449598975246414912863111465364548050391636522567455569326936167362733978208903502309971646876492298053014761680209263210334367950807323938188804032444180691208424432641560933430829642849222471126855975046976264861142200196090873689848421843657191122679730405457613374464333528713595028025499158915631933516131901486678847790154871195366492722123901490942121147791994088903595812840508108723450617408520351474791501477137500457342580798802860226722483071140092607989410381522583658260015263828479974612715845634569484085007653986257235669726683443743485813491066481117864482736645186942197877484560957415923165525303520724765971741826705012467970971652276787231995434582749181129198639619283181022181454009410979226141089115582679800903885255370692759102173591511270474888363142061555783426807681868230735428982849314717966630981691678606125268941949114427742580036471039151646099484825259176902899828540485246941295441813001097452114340209278442506644192595084839750875848820948944425473440101086124918167955646103531186497403430181161785931987860726637908855991154397153654665336597089114095843532096882086827889787605551124993774754527399260647535995679182021376817^2 = 6 $$
answered Mar 22 at 17:49
Will JagyWill Jagy
104k5102201
$endgroup$
ADDED: works pretty well for $y = -p,$ where $p$ is a prime and $p equiv -5 pmod24.$ Not always, though, there is no solution in integers to $u^2 - 499 v^2 = 6,$ so there cannot be a solution to $z^2 - 499^3 x^2 = 6,$
As in comment by Poon,
$$ 62531004125^2 - 6859 cdot 755031379^2 = 6 $$
$$ 30062417279159968032434332377499679191^2 - 6859 cdot 362989027474817934968547501096752545^2 = 6 $$
are the two smallest solutions with $y = -19$
This is a Pell type situation..
Next is $y = -43.$
$$ 8524568252330795722226762076275477^2 - 79507 cdot 30232197004259457995489433581233^2 = 6 $$
$$ 7854330327479105748865242264224583147937033544548000861492635087111452983686976831425073648904157700425958139623153797892698716197^2 - 79507 cdot 1091798726642836794174438412786249442561116671851858490296127998493079755047027851686091414930149044443059275247490273988265975073^2 = 6 $$
$19 pmod 24,$ or $-5 pmod 24$ looks good, jump to $y=-67$
$$ 282978368450687434487209373191993925035994947255483507^2 - 300763 cdot 515989702066780538602608723807499847473549946045669^2 = 6 $$
$$ 110282895315930497927912550392745703024272661444749938273795258966953853545781325097922911621692636895662478761651920330433345436002171835031498514609892669112804144738962046702905275306924383919904704053765669273732362771438086818759096600035555210393331272621715089906882939770617^2 - 300763 cdot 201092537951519476578846016654832921687447975787740751416540268301538352543770725685782163606655569263151546549701485320486846032648563723182340355499316131383039221816513366591020013230115379554131047351811803703438819429350858064042234604275684238200002441987024851219462842279^2 = 6 $$
Alright, $y=-91$ does not work, so it is probably primes $p equiv -5 pmod 24, $
Looks good: $y = -139$ works.
$$4362811041355070169863445353127930955459611243393190431129982455344444937966635765034690725017372595556332974381641720288818463876300492140964198599085172852063910548429431998295054330240877818521750650943853474767971595481380748602727279462883776818584948606778731999374536671484774081257730189572990126422750941506328066671948299736754296627083409459327142431344285253530904973914932476112673827164600483235886537816825709905101946637717937589365141741638476449995905^2 - 2685619 cdot 2662221602563249187278263196011187615830538137637142943172206847128695186087154957324454563072568300849581804322115608951178431618508361010826207991363516489555641531605807855706702830569730399355959940637296610198308891993171988598931443596581366199909035947670242973782535283742571317010626567192123563836486926084664553687222659057469486167968416946690621995773225236487161717059270246838151043190279154072191947271777597482296371378305290636442120248659630720699^2 = 6 $$
$$ 11007229320114200682613076077054270674493469083551395090272400450528009167811973532453164230028506749238868355270609196939003181071165527762688822477732870085553734659733958280600416037711133666637554796387480108123093268249334915679222608686278074350575513887596916117811796252204388381986306317919874745606321543530836091755198128978452469078117805119958799386852835463813171493204121621031603301970267358021486072087211213638671800520653184667508559060289724518769975994491764389949401683257784292501138366997875642534662126840960900976803282849392676436506058878390727089164075298305914748170136999780085836991430800327841415662720459718357825397573172976708908467441^2 - 2685619 cdot 6716697881848674838156030386241616967270051136862684282955991537338414961773579582932757141004204932054213610185226058428383136198936104632942497489498739897183325583340951152435956413801114255489130165449836905051326165370830554673259419604890586134665267687698856933901633921567579748923093505649829262337020641217111875006923069411778584596026715953822127592173142902673636449348881975583202914018304502788869342977433804925999933660151442708630639152564344393622683488924533630738934557423091516094843646237584273637443239810351573953018195234525958355653029554536372430979361344097937950355948103639632777321526741953229033254611279820660546487889155631849994795^2 = 6 $$
$y = -163 $ works. The others should be checked for class number! No, $139$ has class number 3.
$$
1085997992037838057157328137528530814161327411705546203392088990536040293^2 - 4330747 cdot 521852316521211748718181001351655538497525757942306914225856323090563^2 = 6 $$
$$ 50576439676678868925447962668907434500838480927375612223861414693409656592498369002412775393461481995344704306058443464631047373837930992505267663441324334534575755546957820383354011427592442618572553822492422632438018310472471467898207758462496052055942015486413777383860175293508915160725621756472206275639559775926508637417532793267711813496401310961017300099032326961359121416479939614082943062346592497129301169840684667781108633285175838272455967344224725866133380586503707411632734800342621002287656549985786259734994462255794013879381219457810712405084283774220828444140059155951479917295341262660858326466541291848167815166877848998169467570609203031564636509860322954392994510333265978746217370992701911980229349147987494073342207262560127339452333844537013716461592403989956476819621648310312838004247585673708244092703910695114947918019837457181216465905991224195278477515712580521442345725333566315384496982715121821384411921111090497302852848529853789432402185245489658332331342402733989663728321530595214247562396845918152405928944606618632203463997552998841227239534262770964180949700862983087031814190544748296635415256018314980663318640218710377201213304383260086867630769339280626456705440420391795961162063402561211243834067151983^2 - 4330747 cdot 24303389509168264155926836231677339583236279792966852959122818227864577731252374449598975246414912863111465364548050391636522567455569326936167362733978208903502309971646876492298053014761680209263210334367950807323938188804032444180691208424432641560933430829642849222471126855975046976264861142200196090873689848421843657191122679730405457613374464333528713595028025499158915631933516131901486678847790154871195366492722123901490942121147791994088903595812840508108723450617408520351474791501477137500457342580798802860226722483071140092607989410381522583658260015263828479974612715845634569484085007653986257235669726683443743485813491066481117864482736645186942197877484560957415923165525303520724765971741826705012467970971652276787231995434582749181129198639619283181022181454009410979226141089115582679800903885255370692759102173591511270474888363142061555783426807681868230735428982849314717966630981691678606125268941949114427742580036471039151646099484825259176902899828540485246941295441813001097452114340209278442506644192595084839750875848820948944425473440101086124918167955646103531186497403430181161785931987860726637908855991154397153654665336597089114095843532096882086827889787605551124993774754527399260647535995679182021376817^2 = 6 $$
answered Mar 22 at 17:49
Will JagyWill Jagy
104k5102201
edited Mar 23 at 0:02
answered Mar 22 at 17:49
Will JagyWill Jagy
104k5102201
answered Mar 22 at 17:49
Will JagyWill Jagy
104k5102201
answered Mar 22 at 17:49
Will JagyWill Jagy
104k5102201
104k5102201
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1
$begingroup$
You can massively improve time by using the fact that $n=x^2y^3$ iff $n$ is powerful - every prime factor of $n$ appears with exponent at least $2$. Using Sage's built-in
factorfunction I was able to check $|z|leq 10^6$ in under 20 seconds.$endgroup$
– Wojowu
Mar 22 at 16:08
$begingroup$
@Wojowu that was the idea I used, but I implemented the factorization algorithm myself. I'll try that implementation instead. Thanks
$endgroup$
– Erik T.
Mar 22 at 16:37
2
$begingroup$
FWIW, from Mathematica, the smallest value of $|y|$ for which this has a solution seems to be $y=-19$ with $(x,z)=(755031379,62531004125)$, but I am not sure about this.
$endgroup$
– Poon Levi
Mar 22 at 17:19
$begingroup$
@PoonLevi Wow, nice solution, how did you find it?
$endgroup$
– Erik T.
Mar 22 at 17:44
$begingroup$
@ErikT. I just did some manual brute forcing with the
FindInstancefunction$endgroup$
– Poon Levi
Mar 23 at 5:36