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/
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