Find all the Pythagorean triples that form an arithmetic sequence.

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- Nov 1st 2010, 05:20 PMMATNTRNGPythagorean Triples II
Find all the Pythagorean triples that form an arithmetic sequence.

- Nov 1st 2010, 06:02 PMDrexel28
- Nov 1st 2010, 09:23 PMSoroban
Hello, MATNTRNG!

Quote:

Find all the Pythagorean triples that form an arithmetic sequence.

Assume that Pythagorean triples consist of positive integers.

We have: .

Hence: .

. .

. . . .

. . . .

And we have: .

Since , the Pythagorean triples are: .

. . where is any positive multiple of 3.

- Nov 2nd 2010, 07:13 AMmelese
A useful strategy would be to denote the sides and , where and are integers; these form an arithmetic sequence. We assume that .

The following relation holds: . From , we find that . But since is one of the triangle's sides, so we can divide by to get .

Thus Pythagorean triples that form an arithmetic sequence are given by . In particular, this implies that the**only**Pythagorean triple that consists of consecutive postive integers is .