# Integer points on elliptic curve

Artur grafix at csl.pl
Fri Oct 19 09:17:49 CEST 2007

```Dear Seqfans,
I  have question how find all integer solutions of elliptic curve
a^2+/-1689b^2=c^3
another than
{a,b,c}={{41,1,-2},{3370,82,4}
And question for MAGMA experts what mean parameters A in MAGMA  procedure:
Sort([ p[1] : p in IntegralPoints(EllipticCurve([A, 1689])) ]);

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ARTUR

Dean Hickerson pisze:
> David Wilson wrote:
>
>
>> What values can occur in A002445?
>>
>> Could we write a sorted version of A002445?
>>
>> It looks like it might start
>>
>> 1,6,30,42,66,138,282,330,354,498,510,642,690,798,870,1002,...
>>
>> How could we be sure we had them all?
>>
>
> >From the OEIS entry:
>
>     %N A002445 Denominators of Bernoulli numbers B_2n.
>     %C A002445 From the Von Staudt-Clausen theorem, denominator(B_2n) =
>                product of primes p such that (p-1)|2n.
>
> Except for a(0)=1, all terms are divisible by 6 and are squarefree.  To
> test such a number k to see if it's in the sequence, let 2n be the least
> common multiple of all p-1 for which p is a prime divisor of k.  Now list
> the primes p such that p-1 divides 2n.  If all of them are divisors of k,
> then k is in the sequence; otherwise it's not.
>
> For example, consider  k = 78 = 2 * 3 * 13.  The LCM of 2-1, 3-1, and 13-1
> is 12, so 2n=12.  The primes p such that p-1 divides 12 are 2, 3, 5, 7, and
> 13.  Since 5 and 7 aren't divisors of 78, 78 is not in the sequence.
>
> Dean Hickerson
> dean at math.ucdavis.edu
>
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