## Many to one function which maps positive odd integers which are the sum of two squares onto the positive odd integers

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• 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
Message 1 of 3 , Mar 2, 2009
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

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?

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
• ... It is known that for an odd positive integer A, the following two statements are equivalent: - A is a sum of two squares. - In the prime factorization of
Message 2 of 3 , Mar 2, 2009
Kermit Rose wrote:
>
> Which odd positive integers are the sum of two squares?
>
>

It is known that for an odd positive integer A, the
following two statements are equivalent:

- A is a sum of two squares.

- In the prime factorization of A, no prime of the form
4x+3 appears an odd number of times.
• ... Hello Jack. I overlooked that multiplying a sum of squares times a square also yields a sum of squares. Thus I wish to amend my Function that maps the
Message 3 of 3 , Mar 2, 2009
Jack Brennen wrote:
>
> Kermit Rose wrote:
>>
>> Which odd positive integers are the sum of two squares?
>>
>>
>
> It is known that for an odd positive integer A, the
> following two statements are equivalent:
>
> - A is a sum of two squares.
>
> - In the prime factorization of A, no prime of the form
> 4x+3 appears an odd number of times.
>
>
>

Hello Jack.

I overlooked that multiplying a sum of squares times a square also
yields a sum of squares.

Thus I wish to amend my Function that maps the positive odd sums of two
squares onto the odd positive integers.

I had not yet defined to what the function would map squares of primes
equal to 3 mod 4.

I extend the function to cover this case by

F(9) = 3
F(49) = 7

F( q**2) = q if q is a prime equal to 3 mod 4.

Kermit Rose
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