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MathGroup Archive 2006

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Re: Re: Finding the Number of Pythagorean Triples below a bound

  • To: mathgroup at smc.vnet.net
  • Subject: [mg68309] Re: [mg68242] Re: Finding the Number of Pythagorean Triples below a bound
  • From: "Carl K. Woll" <carlw at wolfram.com>
  • Date: Tue, 1 Aug 2006 06:59:32 -0400 (EDT)
  • References: <eaeqa3$53v$1@smc.vnet.net> <200607300848.EAA25171@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

titus_piezas at yahoo.com wrote:
> Hello all,
> 
> To be specific, I am looking for some code applicable to the more
> general bivariate polynomial,
> 
> Poly(a,b) = c^2
> 
> where {a,b} are positive integers, 0 < a <=b, and of which the
> Pythagorean triples are just a special case. The problem is to find
> S(10^m) which is the number of solutions with c < 10^m for as "high" as
> m=5,6 with a reasonable run-time of, say, an hour or less.
> 
> Anybody knows of such code?
> 
> -Titus

As specified, your problem has polynomials which yield an S value of 
Infinity. Besides the trivial example b-a, there are also polynomials 
which gives Pell's equation, e.g.,  b^2-2a^2:

b^2 - 2a^2 == 1

has an infinite number of integer solutions with 0 < a <= b.

Is this intentional? If your problem has other restrictions which force 
the number of solutions to be finite you should tell us, as algorithms 
for the finite case will probably be very different from algorithms for 
the infinite case.

Carl Woll
Wolfram Research


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