# A024407 and related

Antreas P. Hatzipolakis xpolakis at otenet.gr
Sun Jun 11 16:21:46 CEST 2000

```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.

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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,.....
Name: Smallest common area of n Pythagorean triangles (not necessarily
primitive).

How about primitive ones? Well...The sequence now is:

6, 210, 13123110, ......
Name: Smallest common area of n primitive Pythagorean triangles.

Antreas

```