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Pythagorean triples were also used in ancient Egypt. For exam-ple, a rough-and-ready way to produce a right angle is to take a piece of string, mark it into 12 equal segments, tie it into a loop, and hold it taut in the form of a 3-4-5 triangle, as illustrated in Figure 2.2.
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a;b;c) where a2 +b2 = c2. Examples include (3;4;5), (5;12;13), and (8;15;17). Below is an ancient Babylonian tablet listing 15 Pythagorean triples. It is called Plimpton 322 (George Arthur Plimpton donated it to Columbia University).
The study of Pythagorean triples began about a thousand years before the time of Pythagoras(585-447B.C.) since there are Babylonian tablets dating about 1500B.C. con- taining lists of such triples including (3, 4, 5) and (4961, 6480, 8161).
Application 1: Polynomial Pythagorean triples Consider polynomials f(x);g(x);h(x) satisfying f(x)2 + g(x)2 = h(x)2 and all nonzero. Call the triple primitive if (f(x);g(x)) = 1. Theorem The primitive Pythagorean triples in R[x] are given by f(x) = c(u(x)2 v(x)2);
7 Ιουλ 2020 · Finding Pythagorean triples is one of the earliest problems in the theory of numbers, and certainly, Pythagorean triples are some of the oldest known solutions of the nonlinear Diophantus...
Triples of numbers like (5, 12, 13) are called Pythagorean triples. The theorem itself is much more than that. The theorem not only lists a few examples for evidence but states and proves that for all triangles, the relation a2 + b2 = c2 holds if and only if the triangle is a right angle triangle.
It was discovered by the author of this paper that each triple revolves around a unique central square— the cornerstone that allowed this geometrical interpretation of Pythagoras’ and Plato’s families of triples.
