Primitive Pythagorean triple

Definition

A primitive Pythagorean triple is a triple $(a,b,c)$ of positive integers that is a Pythagorean triple (i.e., $a^{2}+b^{2}=c^{2}$) and either of these two equivalent conditions is satisfied:

1. The $a,b,c$ are pairwise coprime integers.
2. The $a,b,c$ are together coprime, i.e., they have no factor common to both of them.

We study primitive Pythagorean triples up to the symmetry of interchanging the first two members -- thus, $(a,b,c)$ and $(b,a,c)$ are considered the same primitive Pythagorean triple.

Definition in terms of rational points on the circle

Primitive Pythagorean triples (not viewed up to the interchange symmetry) are naturally in bijection with the rational points in the first quadrant of the unit circle centered at the origin, via the mapping:

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

When viewed up to the interchange symmetry of $a$ and $b$, each primitive Pythagorean triple corresponds to a pair of points in the first quadrant that are symmetric about the $y=x$ line.

Equivalence of definitions

Conditions (1) and (2) are equivalent for Pythagorean triples because any factor common to two of the three variables $a,b,c$ is also a factor of the third. For instance, if $d$ divides $a$ and $b$, then $d^{2}$ divides $c^{2}$ and hence $d$ divides $c$.

Classification

Further information: classification of primitive Pythagorean triples

Primitive Pythagorean triples (up to the interchange symmetry) are classified by pairs of positive integers $u,v$ with $u<v$, $u$ coprime to $v$, and exactly one of $u$ and $v$ even. The parametrization mapping is:

$\!(u,v)\mapsto (v^{2}-u^{2},2uv,v^{2}+u^{2})$

Particular cases

$u$	$v$	$a$	$b$	$c$
1	2	3	4	5
2	3	5	12	13
1	4	15	8	17
3	4	7	24	25
2	5	21	20	29
4	5	9	40	41
1	6	35	12	37
5	6	11	60	61
2	7	45	28	53
4	7	33	56	65

Facts

In a primitive Pythagorean triple $(a,b,c)$:

• $c$ is always odd.
• One of $a$ and $b$ is even and the other one is odd.
• The bigger of $a$ and $b$ may be either even or odd. In triples such as $(3,4,5)$, $(5,12,13)$, and $(7,24,25)$, the bigger number is even. In triples such as $(8,15,17)$, the smaller number is even.