Find all the Pythagorean triples that form an arithmetic sequence.
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 .