Pythagorean Triples — Common Triples & How to Find Them

A Pythagorean triple is a set of three positive integers $(a,b,c)$ that satisfy $a^2 + b^2 = c^2$. In plain language, the three numbers are whole-number side lengths of a right triangle, and $c$ is the hypotenuse. The classic example is $(3,4,5)$ because $3^2 + 4^2 = 5^2$.

$$a^2 + b^2 = c^2$$

Use the idea only when all three values are positive integers. Many right triangles satisfy the Pythagorean theorem, but only some give integer side lengths.

Common Pythagorean Triples To Know

These show up often enough that they are worth recognizing on sight:

• $(3,4,5)$
• $(5,12,13)$
• $(8,15,17)$
• $(7,24,25)$

Their multiples work too. For example, doubling $(3,4,5)$ gives $(6,8,10)$, and

$$6^2 + 8^2 = 36 + 64 = 100 = 10^2$$

That is why many non-primitive triples are just scaled copies of smaller ones.

What Makes A Triple Primitive

A primitive Pythagorean triple has no common factor greater than $1$. For example, $(3,4,5)$ is primitive, but $(6,8,10)$ is not because all three numbers are divisible by $2$.

This matters because every non-primitive triple comes from scaling a primitive one. If you understand the primitive triples, you understand the larger family too.

How To Find Pythagorean Triples

There are two practical ways to get new ones.

Scale A Triple You Already Know

If $(a,b,c)$ is a Pythagorean triple and $k$ is a positive integer, then $(ka,kb,kc)$ is also a Pythagorean triple because

$$(ka)^2 + (kb)^2 = k^2a^2 + k^2b^2 = k^2(a^2+b^2) = k^2c^2 = (kc)^2$$

This is the fastest way to build examples such as $(9,12,15)$ or $(10,24,26)$.

Use Euclid's Formula

If $m$ and $n$ are integers with $m > n > 0$, then

$$a = m^2 - n^2,\quad b = 2mn,\quad c = m^2 + n^2$$

gives a Pythagorean triple.

If you want a primitive triple, meaning the three numbers have no common factor greater than $1$, you also need $m$ and $n$ to be coprime and not both odd.

Worked Example: Generate A Triple

Take $m = 4$ and $n = 1$. Since $m > n > 0$, Euclid's formula applies.

Then

$$a = 4^2 - 1^2 = 15,\quad b = 2(4)(1) = 8,\quad c = 4^2 + 1^2 = 17$$

So $(8,15,17)$ is a Pythagorean triple.

You can verify it directly:

$$8^2 + 15^2 = 64 + 225 = 289 = 17^2$$

Now scale it by $2$ and you get $(16,30,34)$. The right-triangle shape stays the same, but the side lengths double.

This example shows both main ideas at once: Euclid's formula creates a triple, and scaling creates more.

Common Mistakes With Pythagorean Triples

Forgetting The Whole-Number Condition

The equation $a^2+b^2=c^2$ has many real-number solutions, but a Pythagorean triple requires all three values to be positive integers.

Calling Every Valid Triple Primitive

$(6,8,10)$ is a valid triple, but it is not primitive because all three numbers share a common factor of $2$.

Mixing Up "Triple" And "Primitive Triple"

A triple only needs to satisfy $a^2+b^2=c^2$ with positive integers. The extra conditions on $m$ and $n$ matter only if you want the triple to be primitive.

Putting The Largest Number In The Wrong Place

In a triple $(a,b,c)$, $c$ is the hypotenuse, so it must be the largest number.

When Pythagorean Triples Are Useful

They show up in right-triangle geometry, coordinate geometry, and introductory number theory. They are also useful when you want to check quickly whether three whole numbers can form a right triangle.

In proof-based math, they are a standard example of a Diophantine equation: an equation where you look for integer solutions instead of all real-number solutions.

Try A Similar Problem

Use $m = 5$ and $n = 2$ in Euclid's formula, then verify the result in $a^2 + b^2 = c^2$. If you want one more step, explore the Pythagorean Theorem to see how the same relationship is used to solve for missing side lengths.