Associated Topics || Dr. Math Home || Search Dr. Math

### 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.

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

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search