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A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A triangle whose side lengths are a Pythagorean triple is a right triangle and ...
Berggrens's tree of primitive Pythagorean triples. In mathematics, a tree of primitive Pythagorean triples is a data tree in which each node branches to three subsequent nodes with the infinite set of all nodes giving all (and only) primitive Pythagorean triples without duplication. A Pythagorean triple is a set of three positive integers a, b ...
A Pythagorean triple can be generated using any two positive integers by the following procedures using generalized Fibonacci sequences. For initial positive integers h n and h n+1, if h n + h n+1 = h n+2 and h n+1 + h n+2 = h n+3, then (+ +, +, + + +) is a Pythagorean triple.
A Pythagorean triple has three positive integers a, b, and c, such that a 2 + b 2 = c 2. In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths. [1] Such a triple is commonly written (a, b, c). Some well-known examples are (3, 4, 5) and (5, 12, 13).
There are infinitely many such triples, [19] and methods for generating such triples have been studied in many cultures, beginning with the Babylonians [20] and later ancient Greek, Chinese, and Indian mathematicians. [1] Mathematically, the definition of a Pythagorean triple is a set of three integers (a, b, c) that satisfy the equation [21] a ...
This table lists two of the three numbers in what are now called Pythagorean triples, i.e., integers a, b, and c satisfying a 2 + b 2 = c 2. From a modern perspective, a method for constructing such triples is a significant early achievement, known long before the Greek and Indian mathematicians discovered solutions to this problem.
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2. [ 38] This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3.
A near-isosceles Pythagorean triple is an integer solution to a 2 + b 2 = c 2 where a + 1 = b. The next table shows that splitting the odd number H n into nearly equal halves gives a square triangular number when n is even and a near isosceles Pythagorean triple when n is odd. All solutions arise in this manner.