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What is a Pythagorean triple?

A Pythagorean triple is three whole-number side lengths of a right triangle. The smallest familiar one is 3-4-5, because 9 + 16 = 25. Other primitive examples include 5-12-13, 8-15-17, 7-24-25, and 20-21-29.

Primitive and scaled

A triple is primitive when the greatest common divisor of the three numbers is 1. Multiply every side by the same whole number and you still have a right triangle, but it is a scaled triple. 6-8-10 is 3-4-5 multiplied by 2. 9-12-15 is 3-4-5 multiplied by 3.

How to test one yourself

1. Check that each side is a whole number, or close enough to one that rounding error is the only difference.
2. Confirm the longest side really is the longest, and that the shorter two can form a triangle with it.
3. Check that the squares satisfy a² + b² = c² using those integers, not a rounded decimal approximation.
4. Divide all three numbers by their greatest common divisor. If that divisor is 1, the triple is primitive. If it is larger, the divided numbers are the primitive triple it comes from.

What is not a triple

A right triangle with decimal sides can still satisfy the theorem. 1.5, 2, and 2.5 is a scaled 3-4-5 written with decimals, and it is a right triangle, but it is not an integer triple. The app’s badge appears only for the integer case.

Common mistakes

• Memorizing a short list and missing a valid triple that is not on it. The test above does not depend on a fixed list.
• Calling every right triangle a triple. The sides have to be whole numbers.
• Calling 6-8-10 primitive. The common factor is 2.

How the app does it

After a successful Basic calculation, or after Check a Triangle reports a right triangle, a badge appears only when the detector finds an integer triple. A scaled result includes the smaller triple, for example that 6-8-10 comes from 3-4-5. The check itself is separate: a triangle can be right without being a triple.

Get Pythagorean Theorem Calculator on Google Play

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