prime factors of Pythagorean hypotenuses

The possible hypotenusesMathworldPlanetmath of the Pythagorean trianglesMathworldPlanetmath ( form the infiniteMathworldPlanetmath sequenceMathworldPlanetmath

5, 10, 13, 15, 17, 20, 25, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45,

the mark of which is in the corpus of the integer sequences of  This sequence has the subsequence A002144

5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137,

of the odd Pythagorean primes.

Generally, the hypotenuse c of a Pythagorean triangle (Pythagorean tripleMathworldPlanetmath) may be characterised by being the contraharmonic mean


of some two different integers u and v (as has been shown in the parent entry), but also by the

Theorem.  A positive integer c is the length of the hypotenuse of a Pythagorean triangle if and only if at least one of the prime factorsMathworldPlanetmath of c is of the form 4n+1.

Lemma 1.  All prime factors of the hypotenuse c in a primitive Pythagorean triple are of the form 4n+1.

This can be proved here by making the antithesis that there exists a prime 4n-1 dividing c.  Then also


where a and b are the catheti in the triple.  But 4n-1 is prime also in the ring [i] of the Gaussian integers, whence it must divide at least one of the factors a+ib and a-ib.  Apparently, that would imply that 4n-1 divides both a and b.  This means that the triple (a,b,c) were not primitive, whence the antithesis is wrong and the lemma true. 

Also the converseMathworldPlanetmath is true in the following form:

Lemma 2.  If all prime factors of a positive integer c are of the form 4n+1, then c is the hypotenuse in a Pythagorean triple.  (Especially, any prime 4n+1 is found as the hypotenuse in a primitive Pythagorean triple.)

Proof.  For proving this, one can start from Fermat’s theorem, by which the prime numbersMathworldPlanetmath of such form are sums of two squares (see the's_theorem_on_sums_of_two_squaresTheorem on sums of two squares by Fermat).  Since the sums of two squares form a set closed under multiplicationPlanetmathPlanetmath, now also the product c is a sum of two squares, and similarly is c2, i.e. c is the hypotenuse in a Pythagorean triple. 

Proof of the Theorem.  Suppose that c is the hypotenuse of a Pythagorean triple (a,b,c); dividing the triple members by their greatest common factor we get a primitive triple (a,b,c) where  cc.  By Lemma 1, the prime factors of c, being also prime factors of c, are of the form 4n+1.

On the contrary, let’s suppose that a prime factor p of  c=pd  is of the form 4n+1.  Then Lemma 2 guarantees a Pythagorean triple (r,s,p), whence also (rd,sd,c) is Pythagorean and c thus a hypotenuse. 

Title prime factors of Pythagorean hypotenuses
Canonical name PrimeFactorsOfPythagoreanHypotenuses
Date of creation 2014-01-31 10:48:20
Last modified on 2014-01-31 10:48:20
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 12
Author pahio (2872)
Entry type Theorem
Classification msc 11D09
Classification msc 51M05
Classification msc 11E25