## Perfect right-angled triangles

### Problem 218

Published on Saturday, 22nd November 2008, 01:00 am; Solved by 2024; Difficulty rating: 55%Consider the right angled triangle with sides a=7, b=24 and c=25.
The area of this triangle is 84, which is divisible by the perfect numbers 6 and 28.

Moreover it is a primitive right angled triangle as gcd(a,b)=1 and gcd(b,c)=1.

Also c is a perfect square.

We will call a right angled triangle perfect if

-it is a primitive right angled triangle

-its hypotenuse is a perfect square

We will call a right angled triangle super-perfect if

-it is a perfect right angled triangle and

-its area is a multiple of the perfect numbers 6 and 28.

How many perfect right-angled triangles with c≤10^{16} exist that are not super-perfect?