Fwd: First 11 values of A121387 coincide with apparently unrelated new seq

[Jonathan Post]

From: jonathan post <jvospost2 at yahoo.com>
Date: Jun 22, 2007 12:31 PM
Subject: Re: First 11 values of A121387 coincide with apparently
unrelated new seq
To: franktaw at netscape.net, g_m at mcraefamily.com,
rayjchandler at sbcglobal.net, jvospost3 at gmail.com,
seqfan at ext.jussieu.fr, alford1940 at aol.com, ralf at ark.in-berlin.de
Cc: jvospost2 at yahoo.com, jvospost3 at gmail.com


Franklin T. Adams-Watters is 100% correct.

The open issue being the multiplicity of such
solutions. The table I gave in seqfans through n=41
shows a number of double solutions, i.e. semiprime
Pythagorean triple hypotenuses in two different ways.


What is the first triple solution?

Perhaps I should make a seq of the first k-tuple
solution of semiprime Pythagorean triple hypotenuses.
Or pehaps not.  Thought and feelings?

Best,

Jonathan Vos Post


--- franktaw at netscape.net wrote:

> So the correspondence is NOT a coincidence.
>
> It is well known that Pythagorean triples can be
> parameterized as:
>
> x^2 - y^2
> 2xy
> x^2 + y^2;
>
> the triple is primitive if x and y are relatively
> prime and not both
> odd.
>
> The x^2 + y^2 is the hypotenuse.  It is also well
> known that a number
> is the sum of two squares iff every prime divisor =
> 3 (mod 4) is
> present an even number of times.  So if the x^2 +
> y^2 is a semiprime,
> it is either the product of two primes = 1 (mod 4),
> or the square of a
> prime = 3 (mod 4).  It is not hard to show that the
> latter case is
> exclusively x = p and y = 0, which are not
> relatively prime.  So the
> semiprime Pythagorean triple hypotenuses are exactly
> the products of
> two primes = 1 (mod 4) - hence A121387, once that is
> corrected.
>
> Franklin T. Adams-Watters
>
> -----Original Message-----
> From: g_m at mcraefamily.com <g_m at mcraefamily.com>
>
> Sequence A121387 needs to be corrected to remove
> 329, which, unlike all
> the
> other elements of this sequence, is NOT the product
> of two primes of the
> form 4n+1.
>
>
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