Pythagorean Triples Calculator

A Pythagorean triple is a set of three positive integers a, b, and c that satisfies a squared plus b squared equals c squared. Those values can be the side lengths of a right triangle, with a and b as the legs and c as the hypotenuse. The familiar 3, 4, 5 combination is the smallest example because 9 plus 16 equals 25. This calculator searches all increasing pairs of integer legs and keeps each pair whose calculated hypotenuse is also an integer within the selected limit. A triple is primitive when its three numbers share no common factor greater than one. For example, 3, 4, 5 is primitive, while 6, 8, 10 is not because every value is twice the corresponding value in 3, 4, 5. Non-primitive triples are still perfectly valid right triangles; they are scaled copies of smaller primitive triangles. The calculator labels each result by computing the greatest common divisor of all three sides, making it easy to distinguish fundamental patterns from their multiples. Pythagorean triples have practical and historical importance. Builders can use a 3-4-5 rope or measured layout to establish a right angle. Teachers use triples to demonstrate the theorem without introducing decimal square roots, and number theory uses them to explore divisibility and integer parameterizations. Euclid's formula generates primitive triples from two integers m and n: the legs are m squared minus n squared and two times m times n, while the hypotenuse is m squared plus n squared. Multiplying those values by a positive integer produces related non-primitive triples. The maximum hypotenuse controls the size of the search rather than the number of results. A limit of 15 includes triples with c equal to 5, 10, 13, and 15. Each triangle appears once with the shorter leg first, so swapped duplicates are omitted. The range is capped to keep browser calculations responsive. For larger mathematical investigations, Euclid's formula is more efficient than checking every pair, but this exhaustive method is transparent and guarantees a complete ordered list within the selected range.