KnowraPythagorean triplePythagorean tripleA triple of positive integers satisfying a² + b² = c², the side lengths of a right triangle with integer sides.BriefConnectEuclid's formula: A formula generating Pythagorean triples from two positive integers with opposite parity and no common factor. It constructs every primitive triple by choosing two suitable parameters.Right triangle: A triangle containing one angle of 90 degrees. Each triple gives the three side lengths of a right triangle.Plimpton 322: An Old Babylonian clay tablet listing numerical entries often interpreted as related to right-triangle ratios. Its entries provide early evidence of systematic interest in Pythagorean-triple-like numbers.Carpenter's square: A tool with a right angle used to lay out or check perpendicular lines and corners. A 3-4-5 layout uses a triple to establish a right angle on site.Sum of two squares theorem: A theorem characterizing which primes and integers can be expressed as sums of two integer squares. Every triple expresses c² as a sum of two positive squares, a central theme of the theorem.Primitive Pythagorean triple: A Pythagorean triple whose three entries have greatest common divisor one. Primitive triples are the basic cases from which all others arise by scaling.Diophantine equation: An equation whose solutions of interest are integers or other specified rational numbers. Finding triples is an integer-solution problem for a quadratic equation.Babylonian mathematics: The mathematical practices developed in ancient Mesopotamia, recorded on cuneiform tablets. Some Babylonian records predate Greek proofs and preserve numerical triangle relationships.Surveying: The measurement and mapping of land, boundaries, and physical features. Integer side ratios offer practical constructions for checking perpendicular directions.Fermat's right triangle theorem: The theorem that no positive integer square is the area of a right triangle with integer sides. It shows that integer side lengths impose surprising limits on possible triangle areas.Show all 26Linked from 11 pagesInfinite descentBroader topic: The classic descent constructs a smaller primitive triple while preserving the required square conditions.Coprime integersRelated: Primitive triples are parametrized using two coprime integers of opposite parity.Brahmagupta–Fibonacci identityBroader topic: Combining two square representations can produce new integer square representations and triples.Perfect squareRelated: Each triple gives a perfect square equal to the sum of two other squares.Inverse Pythagorean theoremBroader topic: Such triples give integer-sided examples to which the inverse theorem applies.Show all 11