Hello, MATNTRNG!
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.
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 .