# [seqfan] Re: Diophantine 4+5x^3=y^2 A152463

zak seidov zakseidov at yahoo.com
Fri Dec 5 19:12:07 CET 2008

```Max,
thx a lot, zak

%I A152463
%S A152463 0,1,4,532
%N A152463 Numbers n with property that 4 + 5n^3 is a square.
%C A152463 List of solutions is complete ( Max Alekseyev, Dec 05 2008 (maxale(AT)gmail.com)
%e A152463 4 + 5n^3 = s^2, s = 2,3,18,27438.
%o A152463 (PARI) {for(x=0,2*10^9,if(issquare(4+5*x^3,&y),print(x","y)))}
%K A152463 fini,full,nonn
%O A152463 0,3
%A A152463 Zak Seidov (zakseidov(AT)yahoo.com), Dec 05 2008

--- On Fri, 12/5/08, Max Alekseyev <maxale at gmail.com> wrote:

> From: Max Alekseyev <maxale at gmail.com>
> Subject: [seqfan] Re: Diophantine 4+5x^3=y^2
> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> Cc: "seqfaneu" <seqfan at seqfan.eu>
> Date: Friday, December 5, 2008, 12:10 PM
> Zak,
>
> Multiplying by 5^2 and making substitution
> x' = 5x, y' = 5y
> we got a Mordell curve: y'^2 = x'^3 + 100
> for which we can query
> http://tnt.math.metro-u.ac.jp/simath/MORDELL/
>
> We are interested in solutions divisible by 5, and up to a
> sign there
> are only 4 of them.
> So, your list is complete.
>
> Regards,
> Max
>
> On Fri, Dec 5, 2008 at 8:51 AM, zak seidov
> <zakseidov at yahoo.com> wrote:
> > Dear SeqFans,
> >
> > Diophantine 4+5x^3=y^2 has 4 solutions (x,y >= 0):
> > x=0,1,4,532 and y=2,3,18,27438.
> > Are there more solutions (x>2*10^9)?
> > thx, zak
> >
> > (17:59) gp > {for(x=0,2*10^9,if(issquare(4+5*
> x^3,&y),print(x","y)))}
> > 0,2
> > 1,3
> > 4,18
> > 532,27438
> > (18:46) gp >
> >
> >
> >
> >
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
>
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>
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```