This section in pdf form triples3.pdf
It is well-known that if is a solution to a Pythagorean triangle, where is the hypotenuse, then the Mersenne prime divides , and the Fermat prime divides . In this section, I show that if is a solution to a primitive Pythagorean triangle, where is a non-negative integer, then every Mersenne prime less than or equal to divides , and every Fermat prime less than or equal to divides .
If is a solution to the Pythagorean triangle then is a Pythagorean triple. If, additionally, , and are pairwise relatively prime then is a primitive Pythagorean triple (PPT), and is a Primitive Pythagorean triangle.
All PPT's are given by the parametric equations
When computing the PPT , it is convenient to express , and in terms of Gaussian integers . To do so, let where and are relatively prime, positive integers having opposite parity. Let and . And let be a non-negative integer. Then there exists positive integers and such that
Thus, all primitive Pythagorean triples of the form are given by the parametric equations
Then
If and are positive integers such that is a primitive Pythagorean triangle, and if divides then we are going to show the following:
Previously, it was shown (here), that those three items are true for the case
I will, first, state and prove a theorem on the divisors of , and of , where is a PPT. Then the case where will be proven in the corollary. If in equation (19) then
Which implies
Similarly,
Which implies
Therefore .
where and . Which implies
Similarly
Then, if
Therefore .
Let be a primitive Pythagorean triangle where is a nonnegative integer. Let be any Mersenne prime less than or equal to . And, let be any Fermat prime less than or equal to . Then
is a primitive Pythagorean triangle where is a nonnegative integer then divides .
So there exists integers and , one odd the other even, such that and . Hence divides . Assume true for , then
. Let and . Then if divides , divides .