A perfect number is a positive integer that is equal to the sum of its divisors excluding itself. For example is a perfect number since its divisors are and where All even perfect numbers are of the form where is a prime. Such primes are called Mersenne primes. The first five perfect numbers are
It's not known if there are any odd perfect numbers.
From corollary (1) and theorem (2) we see that it is no coincidence that the area of the triangle, the smallest Pythagorean triangle, is , the smallest perfect number.
Let be a positive integer such that is a prime (Mersenne prime). And let be a primitive Pythagorean triangle. Then, as table (4) shows, the area is a multiple of the perfect number
Examples
Consider the two primitive Pythagorean triangles and In this case
Since is a prime, we know that each area must be a multiple of the perfect number And indeed, and
Consider the two primitive Pythagorean triangles and In this case
Since is a prime, we know that each area must be a multiple of the perfect number And indeed, and
Consider the two primitive Pythagorean triangles and In this case
Since is a prime, we know that each area must be a multiple of the perfect number And indeed, and
Note: Since is also a multiple of each of 6 and 28.
That is,