Diophantine Analysis

Pythagorean Triples

In chapter twelve of his *Brahmasphutasiddhanta*, Brahmagupta provides a formula useful for generating Pythagorean Triples:

12.39. The height of a mountain multiplied by a given multiplier is the distance to a city; it is not erased. When it is divided by the multiplier increased by two it is the leap of one of the two who make the same journey.

Or, in other words, if *d = mx/(x + 2)*, then a traveller who “leaps” vertically upwards a distance *d* from the top of a mountain of height *m*, and then travels in a straight line to a city at a horizontal distance *mx* from the base of the mountain, travels the same distance as one who descends vertically down the mountain and then travels along the horizontal to the city.Stated geometrically, this says that if a right-angled triangle has a base of length *a = mx* and altitude of length *b = m + d*, then the length, *c*, of its hypotenuse is given by *c = m (1+x) – d*. And, indeed, elementary algebraic manipulation shows that *a*^{2} + b^{2} = c^{2} whenever *d* has the value stated. Also, if *m* and *x* are rational, so are *d, a, b* and *c*. A Pythagorean triple can therefore be obtained from *a, b* and *c* by multiplying each of them by the least common multiple of their denominators.

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