[seqfan] Re: Is A002144 identical to "primes in A020882 Ordered hypotenuses (with multiplicity) of primitive Pythagorean triangles?

[Charles Greathouse]

Clearly (working mod 4) the primes in A020882 are all of the form
4n+1, so the only question is whether all such primes are represented.

Fermat's two-square theorem says that all 4n+1 primes can be
represented as a^2 + b^2, so it remains only to show that gcd(b-a,
2ab) = 1.  If b-a is even then a^2 + b^2 is even and so not 1 mod 4.
So suppose there is some prime q dividing both b-a and ab.  Since q is
a prime element, it divides at least one of a and b; WLOG, q | a.
Then since q  | b-a, q | b, so q^2 | a^2 + b^2.  But then a^2 + b^2 is
not prime.

So yes, the primes in A020882 are precisely those of the form 4n+1.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Thu, Sep 9, 2010 at 2:13 AM, Jonathan Post <jvospost3 at gmail.com> wrote:
> Through 353, this is identical to "primes in A020882 Ordered
> hypotenuses (with multiplicity) of primitive Pythagorean triangles.
>
> A002144                  Pythagorean primes: primes of form 4n+1.
> (Formerly M3823 N1566)
>        5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149,
> 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313,
> 317, 337, 349, 353
>
> I look through the b-list of the former, factorizing those with
> multiplicity > 1.
>
> Is there an easy proof that this is so?  That is, if the multiplicity
> is >1 for the hypotenuse H of a primitive Pythagorean triangle, then H
> must be composite?
> The latter statement is true through H = 2329 = 17 * 137 .  Is this
> obvious?  Is it interesting enough to be a comment to either seq?
>
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>