List of Pythagorean Triples

There are 52 Pythagorean triples with hypotenuse 100 or less, and 16 of them are primitive, meaning their three numbers share no common factor. Allowing hypotenuses up to 300 gives 47 primitive triples. The tables below list them all, sorted by hypotenuse, with Euclid's parameters m and n, the perimeter and the area of each right triangle.

New to triples? Start with what a Pythagorean triple is, then come back here to look things up. Every row was produced by a program and then checked against a² + b² = c²; nothing was typed by hand.

How to read the tables

• a, b, c are the side lengths: a is the shorter leg, b the longer leg and c the hypotenuse. Each triangle appears once, so 3-4-5 is listed but 4-3-5 is not.
• Primitive? says whether a, b and c have no common factor. See primitive Pythagorean triples for the test.
• m, n are the two numbers that produce a primitive triple in Euclid's formula: the odd leg is m² − n², the even leg is 2mn, and c = m² + n². The even leg can be either a or b: in 8-15-17 (m = 4, n = 1) the even leg 8 is the shorter one, while in 5-12-13 (m = 3, n = 2) the even leg 12 is the longer one.
• k × primitive appears instead for non-primitive rows, naming the primitive triple it is a multiple of. For example, 9-12-15 is 3 × (3, 4, 5).
• Perimeter is a + b + c. Area is 1 over 2ab, because the two legs are the base and height of a right triangle.

All Pythagorean triples with hypotenuse up to 100

The full list of 52 triples, sorted by hypotenuse and then by the shorter leg. The 16 primitive triples are marked yes; every other row is a whole-number multiple of one of them.

All primitive Pythagorean triples with hypotenuse up to 300

These 47 triples are the building blocks: multiply any row by k = 2, 3, 4, … to get the non-primitive triples above it. For each one, m and n have no common factor and one of them is even, as the theory predicts.

Need a bigger list, or every triple containing a particular number? The Pythagorean triples generator builds custom lists in your browser.

Patterns to look for in the tables

Which numbers can be a hypotenuse

Only 42 different numbers appear in the c column of the first table. The hypotenuses of primitive triples up to 100 are 5, 13, 17, 25, 29, 37, 41, 53, 61, 65, 73, 85, 89 and 97, and every one of them leaves remainder 1 when you divide it by 4. Primes such as 7, 11, 19 and 23, which leave remainder 3, never appear as a hypotenuse at all. The rule is that a number is the hypotenuse of some triple exactly when it has at least one prime factor of the form 4k + 1, such as 5, 13, 17 or 29.

Hypotenuses that repeat

The busiest hypotenuses up to 100 are 65 and 85, with four triples each. For 65 they are 16-63-65, 25-60-65, 33-56-65 and 39-52-65: two primitive triples plus two multiples (5 × 5-12-13 and 13 × 3-4-5). Both 65 = 5 × 13 and 85 = 5 × 17 are products of two different primes of the form 4k + 1, and each extra prime of that kind doubles the number of primitive triples with that hypotenuse.

Families hiding in the rows

• Hypotenuse one more than the longer leg: 3-4-5, 5-12-13, 7-24-25, 9-40-41, 11-60-61, 13-84-85. The shorter leg runs through the odd numbers.
• Hypotenuse two more than the odd leg: 8-15-17, 12-35-37, 16-63-65. Here the even leg goes up in steps of 4.
• Legs that differ by one: 3-4-5, 20-21-29 and 119-120-169. The next is 696-697-985, so these are rare.

Rules every row obeys

• The perimeter is always even. For a primitive triple it equals 2m(m + n): for 20-21-29, 2 × 5 × 7 = 70.
• The area is always a multiple of 6, and the product abc is always a multiple of 60.
• In every primitive row exactly one leg is even, that leg is a multiple of 4, and c is odd.
• Every whole number from 3 upward is a leg of at least one triple; the numbers 1 and 2 never are.

The parabola pattern

Plot each primitive triple as a point, with the odd leg across and the even leg up, and something unexpected appears. The dots are not scattered at random; they line up along smooth curved rows.

Each row is a parabola, and Euclid's formula explains why. Write the point as x = m² − n², y = 2mn. If you keep m fixed and let n change, then n = y over 2m, and substituting gives one parabola; if you keep n fixed and let m change, you get another:

The rows with fixed m bend back toward the even-leg axis and cross it at y = 2m²; the rows with fixed n sweep out to the right. Where several families cross, the dots look like a woven pattern. The empty region outside the dashed quarter circle is no accident either: every point satisfies x² + y² = c², so a triple with c ≤ 5,000 must sit inside the circle of radius 5,000.

How these tables were made

The first table comes from a brute-force search: for every c from 1 to 100 and every a < c, check whether c² − a² is a perfect square. The core of the program is just this:

for (let c = 1; c <= 100; c++)
  for (let a = 1; a < c; a++) {
    const b = Math.sqrt(c * c - a * a);
    if (Number.isInteger(b) && a < b) console.log(a, b, c);
  }

A second, independent method, Euclid's formula with every multiplier k, produced exactly the same 52 triples. Each row's primitivity, m, n, perimeter and area were then computed from the triple and re-checked. Because the search tries every possibility, nothing can be missing from the list.

Want to practise using the tables? The cheat sheet lists the triples most worth memorizing, and the tree of primitive triples shows a completely different way to generate the second table.