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### Unique Decomposition of Pythagorean Primes

```Date: 05/19/2002 at 05:47:06
From: S V Kasargod
Subject: Pythagorean primes

A Pythagorean prime is a natural prime that can be expressed as a sum
of squares of two integers. I have a conjecture that a Pythagorean
prime can be expressed as a sum of two squares in one and only one
way. Is this true? If not, are there any counterexamples? If true, is
there a proof available?
```

```
Date: 05/19/2002 at 14:15:05
From: Doctor Paul
Subject: Re: Pythagorean primes

Hi,

Let me begin by quoting from

http://www.math.niu.edu/~rusin/known-math/93_back/2squares

There are well-known forumlae to calculate the number of ways
a number can be written as the sum of two squares.  Gauss gives
one as a footnote to section 182 of the Disquisitiones
Arithmeticae.  Hardy and Wright have an extensive discussion
of the problem (sections 16.9-10 of the fifth edition).

I prefer Gauss' way of doing it, personally.  Basically, you
write N as

2**k
* (p1 ** 2e1)
* (p2 ** 2e2)
* ...
* (pn ** 2en)
* (q1 ** f1)
* ...
* (qn ** fn),

where the p's are all primes congruent to 3 (mod 4) and the
q's are all primes congruent to 1 (mod 4).  (If p is a prime
congruent to 3 (mod 4) and the highest power of p dividing N
is odd, then N cannot be written as the sum of two squares.
That's why we have "2ei" instead of "ei".)

Gauss' formula for the number of ways N can then be written as
the sum of two squares is

[(f1 + 1) * (f2 + 1) * ... * (fn + 1)] / 2

if at least one of the f's is odd, and

[1 + (f1 + 1) * ... * (fn + 1)] / 2

if they are all even.  In this case, since we want to restrict
our solutions to squares, we are more interested in the latter
case.

Thus

65 = 5 * 13

can be written as the sum of squares in (1 + 1)(1 + 1)/2 = 2
ways, and

65 ** 2 = 5 ** 2 * 13 ** 2 [1 + (2 + 1)(2 + 1)] / 2

= 5

ways.

(The original poster only listed four because one of them
is trivial: 0 ** 2 + 65 ** 2, which does not make a terribly
interesting right triangle.)

To find a square that can be written as the sum of squares 7
ways (including the trivial one), we want

[1 + (f1 + 1) * ... * (fn + 1)] / 2 = 7,

so

(f1 + 1) * ... * (fn + 1) = 13,

where all the f's are even.  13 is prime, so we have f1 = 12.
The smallest number that fits this bill is 5 ** 12, so

5 ** 6 = 15625

should be the smallest number that can serve as the hypoteneuse
of a right triangle in 6 non-trivial ways.

This whole process is easily automated.  I used it to get a
list of the smallest number that can be written as the sum of
squares exactly N ways for N = 1 to 512.  It is also relatively
trivial to explicitly list all ways n can be written as the sum
of squares ways once you have its factorization.

Now back to your question:

The primes that can be expressed as the sum of two squares are
precisely the primes congruent to 1 (mod 4).  In this case, our prime
(call it p) would have only one factor; hence the factorization
above would be p = q1 ** f1, where q1 is the prime p and f1 is 1.
Since f1 = 1 is odd, the number of ways to write p as a sum of two
squares is (f1+1)/2 = 1.

This would verify your claim.  If you want a proof of Gauss' result,
consult the sources mentioned in the link I pasted.

I hope this helps.  Please write back if you'd like to talk about
this some more.

- Doctor Paul, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 05/20/2002 at 01:24:17
From: S V Kasargod
Subject: Pythagorean primes

I would like to thank you for your prompt response. I may back to you
after going through the response.
```

```
Date: 05/20/2002 at 08:37:56
From: Doctor Paul
Subject: Re: Pythagorean primes

- Doctor Paul, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Number Theory
High School Number Theory

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