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

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

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