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Topic: a^2+b^2 = c^2+d^2
Replies: 17   Last Post: May 22, 2012 1:33 AM

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Titus Piezas III

Posts: 281
Registered: 12/13/04
a^2+b^2 = c^2+d^2
Posted: Jul 16, 2006 4:25 AM
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Hello all,

It's been proven by Lehmer (1900) that, let P be the number of
primitive solutions to the Pythagorean triple a^2+b^2 = c^2 with
hypotenuse c less than a bound N. Then P/N = 1/(2pi) as N -> infinity.

The question is this: How about the equation,

a^2+b^2 = c^2+d^2 = z

with a,b,c,d all non-zero. Let p be the number of primitive solutions.
Does the ratio p/Sqrt[z] approach a real constant as z -> infinity? Can
it be expressed as a rational multiple of pi?


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