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Primitive Pythagorean Triple

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A primitive Pythagorean triple is a Pythagorean triple (a,b,c) such that GCD(a,b,c)=1, where GCD is the greatest common divisor. A right triangle whose side lengths give a primitive Pythagorean triple is then known as a primitive right triangle.

Let N(p) count primitive Pythagorean triples with perimeter less than p, counting the two orders of the legs only once. Lehmer (1900) showed that

lim_(p->infty)(N(p))/p=(ln2)/(pi^2)
(1)
=0.070230...
(2)

(OEIS A118858).

See also

Primitive Right Triangle, Pythagorean Triangle, Pythagorean Triple, Right Triangle, Sum of Squares Function

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References

Lehmer, D. N. "Asymptotic Evaluation of Certain Totient Sums." Amer. J. Math. 22, 293-335, 1900.Sloane, N. J. A. Sequence A118858 in "The On-Line Encyclopedia of Integer Sequences."

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Primitive Pythagorean Triple

Cite this as:

Weisstein, Eric W. "Primitive Pythagorean Triple." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PrimitivePythagoreanTriple.html

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