Pythagorean Triples Calculator

A Pythagorean triple is a set of three positive integers a, b, c that satisfy the Pythagorean theorem: a squared plus b squared equals c squared. The integers represent the three sides of a right triangle. The most famous example is 3-4-5 because 9 + 16 = 25. Other common examples are 5-12-13, 8-15-17, and 7-24-25.

A primitive Pythagorean triple has no common factor other than 1 (gcd = 1). Examples include 3-4-5, 5-12-13, and 8-15-17. Non-primitive triples are integer multiples of a primitive triple, such as 6-8-10 (which is 2 times 3-4-5). This calculator labels each triple as primitive or non-primitive in the output table.

Euclid's formula states that for any two positive integers m and n with m greater than n, the numbers a = m squared minus n squared, b = 2mn, and c = m squared plus n squared form a Pythagorean triple. When gcd(m, n) = 1 and m and n have opposite parity, the result is a primitive triple. Example: m = 2, n = 1 gives a = 3, b = 4, c = 5.

There are infinitely many Pythagorean triples. For any integer k and any primitive triple (a, b, c), the scaled triple (ka, kb, kc) is also a valid Pythagorean triple. The count of primitive triples with hypotenuse up to a limit N grows roughly as N divided by (2 pi). For example, there are 16 primitive triples with hypotenuse up to 100.

Yes. 6 squared plus 8 squared = 36 + 64 = 100 = 10 squared. However, 6-8-10 is not primitive because gcd(6, 8, 10) = 2. It is the 3-4-5 triple scaled by a factor of 2. The generator will list it as a non-primitive triple in the output table when the limit is at least 10.

There is no largest Pythagorean triple since they are infinite. For any triple (a, b, c), multiplying by any integer k gives another valid triple (ka, kb, kc). The largest triple in any finite list depends on the upper limit you set. With a hypotenuse limit of 100, the largest is 60-80-100 (= 20 times 3-4-5).

Sort the three numbers from smallest to largest. Call them a, b, c where c is largest. Compute a squared plus b squared. If the result equals c squared, the three numbers form a Pythagorean triple. Example: check 9-40-41: 81 + 1600 = 1681 = 41 squared. Yes, it is a triple. Use the Check mode in this calculator for instant verification with full arithmetic shown.

Every primitive Pythagorean triple can be generated by Euclid's formula with exactly one unique pair (m, n) where m greater than n, gcd(m, n) = 1, and m and n have opposite parity. Every non-primitive triple is an integer multiple of a primitive one. So combining Euclid's formula with integer scaling covers all Pythagorean triples.

The 3-4-5 rule is a practical method for ensuring a right angle in construction. Mark 3 units along one wall, 4 units along the adjacent wall, and verify the diagonal measures exactly 5 units. If it does, the corner is a perfect 90 degree angle. Larger multiples like 6-8-10 or 9-12-15 work equally well for bigger layouts.

By strict definition, Pythagorean triples consist only of positive integers. However, any right triangle with rational side lengths can be scaled to a triple with integer sides. Irrational sides (such as 1, 1, root 2) cannot form a Pythagorean triple.

Consecutive-integer triples have the form (n, n+1, hypotenuse) or consecutive legs like (3, 4, 5) and (20, 21, 29). For odd n, the triple is (n, (n squared minus 1) divided by 2, (n squared plus 1) divided by 2). Example: n = 7 gives 7, 24, 25. These come from Euclid's formula with n = 1 and m varying.

The Generate mode uses a combination of Euclid's formula and integer scaling to find all Pythagorean triples whose hypotenuse is at most the chosen limit. It iterates over all coprime, opposite-parity pairs (m, n) to find primitive triples, then generates all multiples that stay within the limit. The results are sorted by hypotenuse, then by the smaller leg.