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The rule attributed to Pythagoras (c. 570 – c. 495 BC) starts from an odd number and produces a triple with leg and hypotenuse differing by one unit; the rule attributed to Plato (428/427 or 424/423 – 348/347 BC) starts from an even number and produces a triple with leg and hypotenuse differing by two units.
Pythagoras. Pythagoras of Samos[a] (Ancient Greek: Πυθαγόρας; c. 570 – c. 495 BC) [b] was an ancient Ionian Greek philosopher, polymath and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of Plato, Aristotle, and, through them, the West in ...
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.
Pythagoras number. In mathematics, the Pythagoras number or reduced height of a field describes the structure of the set of squares in the field. The Pythagoras number p (K) of a field K is the smallest positive integer p such that every sum of squares in K is a sum of p squares. A Pythagorean field is a field with Pythagoras number 1: that is ...
[8] The largest number that always divides abc is 60. [15] Any odd number of the form 2m+1, where m is an integer and m>1, can be the odd leg of a primitive Pythagorean triple. See almost-isosceles primitive Pythagorean triples section below. However, only even numbers divisible by 4 can be the even leg of a primitive Pythagorean triple.
Michael Stifel published the following method in 1544. [3] [4] Consider the sequence of mixed numbers,,,, … with = + +.To calculate a Pythagorean triple, take any term of this sequence and convert it to an improper fraction (for mixed number , the corresponding improper fraction is ).
Pythagorean trigonometric identity. The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions. The identity is.
A Pythagorean prime is a prime number of the form . Pythagorean primes are exactly the odd prime numbers that are the sum of two squares; this characterization is Fermat's theorem on sums of two squares. Equivalently, by the Pythagorean theorem, they are the odd prime numbers for which is the length of the hypotenuse of a right triangle with ...