# Pythagorean bliss

Let $$p$$ be a prime number of the form $$p = 4n + 1$$ ($$n \in \mathbb N$$).

How many different right triangles exist with integer lengths for their legs, such that the hypotenuse is $$p$$?

That is, how many unordered pairs of positive integers $$\{a, b\}$$ exist such that $$a^2 + b^2 = p^2$$?

Example

If $$p = 17$$, there is only one pair: $$\{8, 15\}$$, because the 8-15-17 triangle is the only kind of right triangle with integer sides and hypotenuse equal to 17.

On the other hand, for $$p = 65$$ we have two options: a 16-63-65 triangle and a 33-56-65 triangle. However, this does not count because 65 is not a prime number.

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