Pythagorean prime

A Pythagorean prime p is a prime numberMathworldPlanetmath of the form 4n+1. The first few are 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, etc., listed in A002144 of Sloane’s OEIS. Because of its form, a Pythagorean prime is the sum of two squares, e.g., 29 = 25 + 4. In fact, with the exception of 2, these are the only primes that can be represented as the sum of two squares (thus, in Waring’s problem, all other primes require three or four squares).

Though Pythagorean primes are primes on the line of real integers, they are not Gaussian primesMathworldPlanetmath in the complex planeMathworldPlanetmath. Expressing a Pythagorean prime as a2+b2 (it doesn’t matter whether a<b or viceversa) leads to the complex factorization by simple plugging in of the values thus: p=(a+bi)(a-bi), where i is the imaginary unitMathworldPlanetmath.

Title Pythagorean prime
Canonical name PythagoreanPrime
Date of creation 2013-03-22 16:53:53
Last modified on 2013-03-22 16:53:53
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 4
Author PrimeFan (13766)
Entry type Definition
Classification msc 11A41