Pythagorean triplet

A Pythagorean triplet is a set $\{a, b, c\}$ of three positive integers such that$$ a^2 + b^2 = c^2.$$

That is, $\{a, b, c\}$ is a Pythagorean triplet if there exists a right triangle whose sides have lengths $a$ , $b$ , and $c$ , respectively. For example, $\{3, 4, 5\}$ is a Pythagorean triplet. Given one Pythagorean triplet $\{a, b, c\}$ , we can produce another by multiplying $a$ , $b$ , and $c$ by the same factor $k$ . It follows that there are countably many Pythagorean triplets.

Primitive Pythagorean triplets

A Pythagorean triplet is primitive if its elements are coprimes. All primitive Pythagorean triplets are given by

where the seed numbers $m$ and $n$ are any two coprime positive integers, one odd and one even, such tht $m > n$ . The equations (1) give all Pythagorean triplets, if one presumes of the positive integers $m$ and $n$ only that $m > n$ .

Note 1. Among the primitive Pythagorean triples, the odd cathetus $a$ may attain all odd values except 1 (set e.g. $m := n\!+\!1$ ) and the even cathetus $b$ all values divisible by 4 (set $n := 1$ ).

Note 2. In the primitive triples, the hypothenuses $c$ are always odd. All possible Pythagorean hypotenuses are contraharmonic means of two different integers (and conversely).

Note 3. N.B. that any triplet (1) is obtained from the square of a Gaussian integer $(m\!+\!in)^2$ as its real part, imaginary part and absolute value.

Note 4. The equations (1) imply that the sum of a cathetus and the hypotenuse is always a perfect square or a double perfect square.

Note 5. One can form the sequence (cf. Sloane's A100686)$$ 1,\,2,\,3,\,4,\,7,\,24,\,527,\,336,\,164833,\,354144,\,...$$ taking first the seed numbers 1 and 2 which give the legs 3 and 4, taking these as new seed numbers which give the legs 7 and 24, and so on.