Pythagorean Triples

A Pythagorean Triple is a set of three positive integers namely $a, b$ and $c$ that represent the sides of a right triangle such that the equation ${a^2} + {b^2} = {c^2}$ which is based on the Pythagorean Theorem is satisfied.

We can informally describe the equation of a Pythagorean Triple as:

$${a^2} + {b^2} = {c^2}$$ The sum of the squares of the two smaller positive integers is equal to the square of the largest positive integer.

The geometric interpretation of a Pythagorean Triple can be illustrated by a right triangle with sides $a, b$ and $c$.

• The sides $a$ and $b$ are the legs of a right triangle while $c$ is the hypotenuse, the longest side of a right triangle.
• In addition, side $c$ (hypotenuse) is opposite the 90-degree angle of the right triangle. Sides $a$ and $b$ can be any of two sides of a right triangle that are adjacent to side $c$.

Putting it together, we have the right triangle ABC and the equation that it must satisfy.

From a right triangle perspective, a Pythagorean Triple can be interpreted as the sum of the squares of the shorter sides (legs) equals the square of the longest side (hypotenuse).

One important point, there is no rule or agreement among mathematicians which one of the two legs, $a$ or $b$, is longer than the other.

The legs of a right triangle $a$ and $b$ can either be $a > b$ or $a < b$ which implies that $a$ is longer than $b$, or $a$ is shorter than $b$, respectively.

However, let’s agree only in this lesson that side $a$ is the shorter leg which forces side $b$ to be the longer one. Therefore, $a < b$. It just makes sense since $a$ comes before $b$ in the alphabet.

Note: I intentionally leave the case where the legs of the right triangle are equal because no Pythagorean Triple exists when $a=b$.

Two Types or Kinds of Pythagorean Triple

Primitive Pythagorean Triple (known as “reduced triples”) is a set of three positive integers $a$, $b$, and $c$ with a GCF of $1$. Plus, these three integers must satisfy the equation ${a^2} + {b^2} = {c^2}$.

Examples of Primitive Pythagorean Triples

• $3$, $4$, and $5$ is a Primitive Pythagorean Triple because their GCF is $1$ and they satisfy the equation ${a^2} + {b^2} = {c^2}$.

$${a^2} + {b^2} = {c^2}$$ $${3^2} + {4^2} = {5^2}$$ $${9} + {15} = {25}$$ $${25} = {25}$$
• $5$, $12$, and $13$ is also a Primitive Pythagorean Triple since their largest common divisor is $1$ which means that they are relatively prime. And, it also satisfies the required equation.

$${a^2} + {b^2} = {c^2}$$ $${5^2} + {12^2} = {13^2}$$ $${25} + {144} = {169}$$ $${169} = {169}$$

Non-primitive Pythagorean Triple (known as imprimitive Pythagorean Triple) is a Pythagorean Triple whose three sides of a right triangle namely: $a$, $b$ and $c$ have a GCF larger than $1$. Or, the triplet $a$, $b$ and $c$ have a common factor other than $1$.

Examples of Non-primitive Pythagorean Triples

• Let’s examine the triple $(6, 8, 10)$. The first thing that I observe is that the Greatest Common Factor is $2$. It means that the three numbers have a common factor other than $1$. If this triplet of integers satisfies the equation ${a^2} + {b^2} = {c^2}$, then it must be a Pythagorean triple that is non-primitive.

$${a^2} + {b^2} = {c^2}$$ $${6^2} + {8^2} = {10^2}$$ $${36} + {64} = {100}$$ $${100} = {100}$$

Yes, it checks! Therefore, $(6, 8, 10)$ is a non-primitive Pythagorean triple.

• How about the triple $\left( {32,60,68} \right)$? By quick inspection, all numbers are even and thus they are divisible by 2. Since there is a divisor other than 1, this is a possible non-primitive Pythagorean triple. Check the values with the formula and indeed it works. Therefore $\left( {32,60,68} \right)$ is an example of a non-primitive Pythagorean triple.

$${a^2} + {b^2} = {c^2}$$ $${32^2} + {60^2} = {68^2}$$ $${1,024} + {3,600} = {4,624}$$ $${4,624} = {4,624}$$

That’s it! Remember, if a triplet of numbers satifies the equation ${a^2} + {b^2} = {c^2}$ then it must either be a primitive or non-primitive Pythagorean triple.

Related lessons that might interest you:

List of Pythagorean Triples