# Pythagorean Triples

## Right Triangles and Integer Solutions

A Pythagorean triple is a triple of positive integers

$$
(a,b,c)
$$

satisfying the equation

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

This equation arises from the Pythagorean theorem in Euclidean geometry. If a right triangle has legs of lengths $a$ and $b$, and hypotenuse $c$, then

$$
a^2+b^2=c^2
$$

A classical example is

$$
(3,4,5),
$$

since

$$
3^2+4^2=9+16=25=5^2.
$$

Other examples include

$$
(5,12,13),\qquad (8,15,17),\qquad (7,24,25).
$$

The study of Pythagorean triples is one of the oldest parts of number theory and leads naturally to deeper questions about Diophantine equations.

## Primitive Pythagorean Triples

A Pythagorean triple is called primitive if

$$
\gcd(a,b,c)=1.
$$

For example,

$$
(3,4,5)
$$

is primitive, while

$$
(6,8,10)
$$

is not, since all three numbers are divisible by $2$.

Every nonprimitive triple can be obtained by multiplying a primitive triple by a positive integer. Thus the main problem is to classify primitive triples.

## Basic Properties

Suppose

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

is a primitive Pythagorean triple.

First, $a$ and $b$ cannot both be even, since then $c$ would also be even. This would contradict primitiveness.

They also cannot both be odd. Indeed, if $a$ and $b$ are odd, then

$$
a^2\equiv b^2\equiv 1\pmod 4,
$$

so

$$
a^2+b^2\equiv 2\pmod 4.
$$

But no square is congruent to $2\pmod 4$. Therefore one of $a$ and $b$ must be even, and the other odd.

Without loss of generality, assume $a$ is odd and $b$ is even.

## Parametrization of Primitive Triples

The complete description of primitive Pythagorean triples is given by the following theorem.

**Theorem.** Every primitive Pythagorean triple can be written uniquely in the form

$$
a=m^2-n^2,
$$

$$
b=2mn,
$$

$$
c=m^2+n^2,
$$

where

$$
m>n>0,
$$

$$
\gcd(m,n)=1,
$$

and $m,n$ have opposite parity.

Conversely, every pair $m,n$ satisfying these conditions produces a primitive Pythagorean triple.

Substituting into the equation gives

$$
(m^2-n^2)^2+(2mn)^2
$$

$$
=m^4-2m^2n^2+n^4+4m^2n^2
$$

$$
=m^4+2m^2n^2+n^4
$$

$$
=(m^2+n^2)^2.
$$

Hence the formula always satisfies the Pythagorean equation.

## Examples

Let

$$
m=2,\qquad n=1.
$$

Then

$$
a=2^2-1^2=3,
$$

$$
b=2(2)(1)=4,
$$

$$
c=2^2+1^2=5.
$$

Thus we recover the triple

$$
(3,4,5).
$$

Now let

$$
m=3,\qquad n=2.
$$

Then

$$
a=9-4=5,
$$

$$
b=12,
$$

$$
c=13.
$$

This gives

$$
(5,12,13).
$$

If we multiply by $2$, we obtain the nonprimitive triple

$$
(10,24,26).
$$

## Geometric Interpretation

The equation

$$
x^2+y^2=1
$$

defines the unit circle. Rational points on this circle correspond directly to Pythagorean triples.

Suppose

$$
\left(\frac{a}{c},\frac{b}{c}\right)
$$

lies on the unit circle. Then

$$
\left(\frac{a}{c}\right)^2+\left(\frac{b}{c}\right)^2=1,
$$

which is equivalent to

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

The parametrization of primitive triples arises from describing all rational points on the unit circle by lines through the point

$$
(-1,0).
$$

This connection between geometry and arithmetic is fundamental in modern number theory and arithmetic geometry.

## Historical Remarks

Pythagorean triples were known long before the Greek mathematicians. Babylonian tablets contain examples dating back nearly four thousand years. The systematic study of these triples eventually led to broader investigations of polynomial equations in integers.

The equation

$$
x^n+y^n=z^n
$$

for powers $n>2$ became the subject of entity["historical_event","Fermat’s Last Theorem","proof completed by Andrew Wiles in 1994"]. Unlike the quadratic case, higher powers admit no nontrivial integer solutions. This contrast illustrates the special algebraic structure of quadratic equations.