Re: Sum of first, second, and third powers are all squares

On Mar 9, 7:45=A0pm, Brian S <brianscsm...@[EMAIL PROTECTED]
> wrote:
> I recently came across this interesting pair of numbers: 345 and 184.
> The sum of their first, second, and third powers are all perfect
> squares:
> 345 + 184 =3D 23^2
> 345^2 + 184^2 =3D 391^2
> 345^3 + 184^3 =3D 6877^2
>
> Are there any other nontrivial pairs like this, other than multiples
> of these pairs? =A0I did a brute force search up to 46000 and found only
> this pair and its multiples.

There are infinitely many solutions to this in positive integers which
are not multiples of the pair you found. I believe this was first
proved by Andrew Bremner in a paper in the Internat. J. Math. Math.
Sci. (1986) -- you might be able to find the paper from this link :

http://www.hindawi.com/GetArticle.aspx?doi=3D10.1155/S0161171286000522

The solutions correspond to rational points on a particular elliptic
curve. There are, in fact, no solutions with coprime integers (note
that 23 divides 345 and 184), but this is not easy to prove!

Gaew