Pythagorean Triple Calculator

Find Triples (A, B, C) Where C = nA + k

Pythagorean triples are sets of numbers (A, B, C) in which

A² + B² = C², or equivalently
sqrt(A² + B²) = C

Pythagorean triples represent right triangles with legs of length A and B and a hypotenuse of length C. The first few Pythagorean Triples are

(3, 4, 5)
(5, 12, 13)
(8, 15, 17)
(7, 24, 25)
(20, 21, 29)

These examples are primitive Pythagorean triples because the three numbers do not share any common factors. The set (14, 48, 50) is Pythagorean Triple, but it is not primitive because you can divide each element by 2, yielding the primitive triple (7, 24, 25). Every non-primitive set can be reduced to a primitive set.

For given integers n and k, you can often (but not always) find Pythagorean triangles where C = nA + k. For example, if you choose n = 2 and k = -1, then the first several Pythagorean triples that satisfy the linear relation C = 2A - 1 are:

(3, 4, 5)
(33, 56, 65)
(451, 780, 901)
(6273, 10864, 12545)
(87363, 151316, 174725)
(1216801, 2107560, 2433601)

In these triangles, the ratio of C to A approaches 2, and therefore the ratio of B to A approaches sqrt(3). This means the ratios B/A provide better and better approximations to sqrt(3). Observe,

sqrt(3) = 1.732050807..., and
2107560/1216801 = 1.732049858...

If we choose n = 2 and k = 1, then the first six triples for which C = 2A + 1 are

(8, 15, 17)
(120, 209, 241)
(1680, 2911, 3361)
(23408, 40545, 46817)
(326040, 564719, 652081)
(4541160, 7865521, 9082321)

In these sets, as the ratio of B to A also approaches sqrt(3) since the ratio of C to A approaches 2. The fractions B/A also provide better and better approximations to sqrt(3):

sqrt(3) = 1.732050807..., and
7865521/4541160 = 1.732051061...

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