Math
Pythagorean Triples Calculator
A Pythagorean triple is three whole numbers with a² + b² = c².
This calculator lists every triple up to a hypotenuse limit (with primitive triples marked), checks whether three numbers you enter form a triple and shows the Euclid parameters that generate it, or scales a known triple by any whole-number factor.
Whole number from 5 to 2000.
Try an example
Result
Result
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- Count
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- Detail
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- Euclid's formula
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What a Pythagorean triple is
Three positive whole numbers a, b, c with a² + b² = c² form a Pythagorean triple — they are the side lengths of a right triangle with no square roots involved. A triple is primitive when a, b, and c share no common factor; every other triple is a whole-number multiple of a primitive one.
Euclid's formula gives a primitive triple exactly when m and n are coprime and one of them is even. Multiplying the result by any k gives the non-primitive triples.
| m | n | a = m² − n² | b = 2mn | c = m² + n² |
|---|---|---|---|---|
| 2 | 1 | 3 | 4 | 5 |
| 3 | 2 | 5 | 12 | 13 |
| 4 | 1 | 15 | 8 | 17 |
| 4 | 3 | 7 | 24 | 25 |
| 5 | 2 | 21 | 20 | 29 |
| 5 | 4 | 9 | 40 | 41 |
How to use it
- Pick Generate, Check, or Scale.
- For Generate, set the largest hypotenuse you want and whether to include multiples.
- For Check, enter three whole numbers in any order; the calculator sorts them and tests the largest as the hypotenuse.
- For Scale, enter a triple and the whole-number factor.
How to read the answer
In the generated list, triples without a star are primitive; a star marks a multiple such as 6-8-10. A Yes on the check includes the Euclid parameters m and n that produce the primitive base triple, and the common factor when the triple is a multiple.
Common mistakes and edge cases
- Decimals or square roots are not triples. 1, 1, √2 is a right triangle but not a Pythagorean triple.
- Forgetting that order does not matter: 13, 5, 12 is the same triple as 5, 12, 13. The largest number must be the hypotenuse.
- Assuming every right triangle has whole-number sides. Most do not.
- Very large limits produce long lists; the page caps the limit and the number displayed.
Worked examples
All triples up to 50
Result
(3, 4, 5), (6, 8, 10)*, (5, 12, 13), (9, 12, 15)*, (8, 15, 17), (12, 16, 20)*, (7, 24, 25), (15, 20, 25)*, (10, 24, 26)*, (20, 21, 29), (18, 24, 30)*, (16, 30, 34)*, (21, 28, 35)*, (12, 35, 37), (15, 36, 39)*, (24, 32, 40)*, (9, 40, 41), (27, 36, 45)*, (14, 48, 50)*, (30, 40, 50)*
Primitive triples up to 100
Result
(3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (20, 21, 29), (12, 35, 37), (9, 40, 41), (28, 45, 53), (11, 60, 61), (16, 63, 65), (33, 56, 65), (48, 55, 73), (13, 84, 85), (36, 77, 85), (39, 80, 89), (65, 72, 97)
Check 5, 12, 13
Primitive, m = 3, n = 2
Result
Yes — (5, 12, 13) is a primitive Pythagorean triple
Check 6, 8, 10
Multiple of 3-4-5
Result
Yes — (6, 8, 10) is a Pythagorean triple
Check 4, 5, 6
Not a triple
Result
No — (4, 5, 6) is not a Pythagorean triple
Scale 3-4-5 by 7
Result
(21, 28, 35)
Scale a non-triple
Result
—
Limit too small
Result
—
Frequently asked questions
What are the first Pythagorean triples?+
3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29, 9-40-41, 12-35-37, 11-60-61. Multiples like 6-8-10 and 9-12-15 also work.
What is a primitive Pythagorean triple?+
One where a, b, and c have no common factor. 3-4-5 is primitive; 6-8-10 is not because every side is divisible by 2.
How does Euclid's formula work?+
Pick whole numbers m > n > 0. Then a = m² − n², b = 2mn, c = m² + n² always satisfy a² + b² = c². The triple is primitive when m and n are coprime and not both odd.
Are there infinitely many Pythagorean triples?+
Yes. Euclid's formula produces a new primitive triple for every valid (m, n) pair, and each has infinitely many multiples.
Is 7, 24, 25 a Pythagorean triple?+
Yes: 49 + 576 = 625 = 25². It comes from m = 4, n = 3.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026