# A024407 and related

Antreas P. Hatzipolakis xpolakis at otenet.gr
Mon Jun 12 22:15:23 CEST 2000

```I wrote:

> ID Number: A024407
> Sequence:  210,2730,7980,71610,85470,106260,114114,234780,341880,
>            420420,499590,1563660,1647030,1857240,2042040,3423420,
>            3666390,6587490,7393470,8514660,9279270,12766110,13123110,
>            17957940,18820830,23393370,23573550,29099070
> Name:      Areas of more than one primitive Pythagorean triangle.
>
> _____________________________________________________________________________
>
> It was shown by Fermat that, for any number k, it is possible to find
> k distinct right triangles, each with integral sides, having the same area.
> Find sets for k = 1,2,3,4,5, having as common areas
>
>         6, 210, 840, 341880 = 2^3 * 3 * 5 * 7 * 11 * 37
>
> and
>         37383746400 = 2^5 * 3^3 * 5^2 * 7^2 * 11 * 13^2 * 19,
>
> respectively. Are these the smallest possible values?
> (The American Math. Monthly 45 (1938) p. 248, #E327 by Philip Franklin)
>
> The answer is yes: they are the smallest posiible. So we have the sequence:
>
> 6, 210, 840, 341880, 37383746400,.....
^^^^^^^^^^^
Actually, the 5th term is: 6913932480

Thanks to David W. Wilson for the correction.

Antreas

```