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Problem 218


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?