Problem 218
Perfect right-angled triangles
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≤1016 exist that are not super-perfect?
完美直角三角形
考虑边长为a=7、b=24以及c=25的直角三角形。这个三角形的面积是84，能够被完全数6和28整除。
此外，它还是一个原始直角三角形，因为gcd(a,b)=1且gcd(b,c)=1。
同时，c还是一个完全平方数。
我们称一个直角三角形是完美的，如果
- 它是一个原始直角三角形
- 它的斜边是一个完全平方数
我们称一个直角三角形是超级完美的，如果
- 它是一个完美的直角三角形，而且
- 它的面积是完全数6和28的倍数
在斜边c≤1016的完美直角三角形中，有多少个不是超级完美的？