A positive odd prime integer which is equal to 1 mod 4,

is uniquely represented as the sum of two squares.

Let p be the name of a prime equal to 1 mod 4.

Let r and s be the names of the unique positive integers such that

p = r**2 + s**2.

Define F(p) = r + s.

Extend this mapping from the set of primes equal to 1 mod 4, to

all products of primes equal to 1 mod 4 by

F(p1**a1 p2**a2 . . . p_k**a_k) = F(p1)**a1 F(p2)**a2 F(p3)**a3 .

. . F(p_k)**a_k

F(5) = 2 + 1 = 3

F(13) = 2 + 3 = 5

F(17) = 4 + 1 = 5

F(25) = F(5*5) = F(5)*F(5) = 3 * 3 = 9

etc

Which odd positive integers are the sum of two squares?

We can't say much about this.

One of the things that we can say is that

If z is an odd positive integer which is the sum of two squares,

then one of the squares is odd, and one is even.

z = ( n-m)**2 + (1 + n + m)**2 = n**2 - 2 n * m + m**2 + 1 + n**2 +

m**2 + 2 * n + 2 * m + 2 * n * m

z = 2* n**2 +2 * m**2 + 2 * n + 2 * m + 1

z = 1 + 4 * sum of two distinct triangular numbers.

Where one of the triangular numbers is permitted to be equal to zero.

Does anyone know of (or able to discover) any research previously

done on this function which maps sums of two opposite parity squares

onto the odd positive integers ?

Kermit Rose