The Complete Guide to Pythagorean Triples: Formulas, Generation & List

A Pythagorean Triple consists of three positive integers (a, b, c) that satisfy the Pythagorean equation:

a² + b² = c²

When all three sides of a right triangle are whole numbers, measuring, constructing, and calculating dimensions becomes remarkably clean.

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Primitive vs. Non-Primitive Triples

• Primitive Pythagorean Triple: A triple where a, b, and c share no common factor other than 1 (gcd(a, b, c) = 1). Examples include (3, 4, 5), (5, 12, 13), and (8, 15, 17).
• Non-Primitive Triple: Any integer multiple k × (a, b, c) of a primitive triple. For example, multiplying (3, 4, 5) by k = 2 gives (6, 8, 10), and by k = 10 gives (30, 40, 50). Both satisfy a² + b² = c², but their greatest common divisor is k > 1.

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Euclid’s Formula for Generating Triples

Around 300 BC, Euclid proved in his Elements that every primitive Pythagorean triple can be generated from an arbitrary pair of positive integers m and n, provided:

1. m > n > 0
2. m and n are coprime (gcd(m, n) = 1)
3. One number is even and the other is odd (m - n is odd)

The sides are given by:

a = m² - n²
b = 2mn
c = m² + n²

Verification of Euclid’s Formula:

Let us expand a² + b²:

a² = (m² - n²)² = m⁴ - 2m²n² + n⁴
b² = (2mn)² = 4m²n²

Summing both:

a² + b² = m⁴ + 2m²n² + n⁴ = (m² + n²)² = c²  ■

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Table of the Top 16 Primitive Pythagorean Triples

#	m	n	Leg a	Leg b	Hypotenuse c	Perimeter	Area
1	2	1	3	4	5	12	6
2	3	2	5	12	13	30	30
3	4	1	15	8	17	40	60
4	4	3	7	24	25	56	84
5	5	2	21	20	29	70	210
6	5	4	9	40	41	90	180
7	6	1	35	12	37	84	210
8	6	5	11	60	61	132	330
9	7	2	45	28	53	126	630
10	7	4	33	56	65	154	924
11	7	6	13	84	85	182	546
12	8	1	63	16	65	144	504
13	8	3	55	48	73	176	1320
14	8	5	39	80	89	208	1560
15	8	7	15	112	113	240	840
16	9	2	77	36	85	198	1386

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Surprising Mathematical Properties of Pythagorean Triples

1. Divisibility Rules: In every primitive triple:
   • Exactly one leg is a multiple of 3.
   • Exactly one leg is a multiple of 4.
   • Exactly one side (a, b, or c) is a multiple of 5.
   • The product a × b × c is always divisible by 60 (3 × 4 × 5).
2. Inradius is Always an Integer: The inradius of a right triangle with integer sides is given by: r = (a + b - c) / 2 = n(m - n) Because m and n are integers, the inradius is always a whole number!
3. Odd Hypotenuse: The hypotenuse c of any primitive Pythagorean triple is always an odd number, and every prime divisor of c is of the form 4k + 1.

Test any triple instantly with our Pythagorean Triples Explorer.