prime factors of Pythagorean hypotenuses
The possible hypotenuses of thePythagorean triangles
(http://planetmath.org/PythagoreanTriangle) form theinfinite
sequence
the mark of which ishttp://oeis.org/search?q=a009003&language=english&go=SearchA009003 in the corpus of the integersequences of http://oeis.org/OEIS. This sequence has the subsequence A002144
of the odd Pythagorean primes.
Generally, the hypotenuse of a Pythagorean triangle(Pythagorean triple) may be characterised by being thecontraharmonic mean
of some two different integers and (as has been shownin the parent entry), but also by the
Theorem. A positive integer is the length of thehypotenuse of a Pythagorean triangle if and only if at leastone of the prime factors of is of the form .
Lemma 1. All prime factors of the hypotenuse ina primitive Pythagorean triple are of the form .
This can be proved here by making the antithesis that thereexists a prime dividing . Then also
where and are the catheti in the triple. But is prime also in the ring of the Gaussianintegers, whence it must divide at least one of the factors and . Apparently, that would imply that divides both and . This means that thetriple were not primitive, whencethe antithesis is wrong and the lemma true.
Also the converse is true in the following form:
Lemma 2. If all prime factors of a positive integer are of the form , then is the hypotenuse ina Pythagorean triple. (Especially, any prime isfound as the hypotenuse in a primitive Pythagorean triple.)
Proof. For proving this, one can start from Fermat’stheorem, by which the prime numbers of such form are sums oftwo squares (see thehttp://en.wikipedia.org/wiki/Proofs_of_Fermat's_theorem_on_sums_of_two_squaresTheorem on sums of two squares by Fermat). Since the sums of two squares form a set closed undermultiplication
, now also the product is a sum of twosquares, and similarly is , i.e. is the hypotenusein a Pythagorean triple.
Proof of the Theorem. Suppose that is thehypotenuse of a Pythagorean triple ; dividing thetriple members by their greatest common factor we get aprimitive triple where . ByLemma 1, the prime factors of , being also prime factors of, are of the form .
On the contrary, let’s suppose that a prime factor of is of the form . Then Lemma 2 guaranteesa Pythagorean triple , whence also isPythagorean and thus a hypotenuse.