Pythagorean triplet

A Pythagorean triplet is a set $\{a,b,c\}$ of three positiveintegers such that

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

That is, $\{a,b,c\}$ is a Pythagorean triplet if there exists aright 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 bymultiplying $a$, $b$, and $c$ by the same factor $k$. It follows thatthere are countably many Pythagorean triplets.

Primitive Pythagorean triplets

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

$\{\begin{array}{cc}a=2mn,\hfill & \\ b=m^2-n^2,\hfill & \\ c=m^2+n^2,\hfill & \end{array}$(1)

where the seed numbers $m$ and $n$ are any two coprime positiveintegers, one odd and one even, such tht $m>n$.  If one presumes of the positive integers $m$ and $n$ only that  $m>n$, one obtains also many non-primitive triplets, but not e.g. $(6,\mathrm{\hspace{0.17em}8},\mathrm{\hspace{0.17em}10})$.  For getting all, one needs to multiply the right hand sides of (1) by an additional integer parametre $q$.

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 http://www.research.att.com/ njas/sequences/?q=A100686&language=english&go=SearchA100686)

$$1,\mathrm{\hspace{0.17em}2},\mathrm{\hspace{0.17em}3},\mathrm{\hspace{0.17em}4},\mathrm{\hspace{0.17em}7},\mathrm{\hspace{0.17em}24},\mathrm{\hspace{0.17em}527},\mathrm{\hspace{0.17em}336},\mathrm{\hspace{0.17em}164833},\mathrm{\hspace{0.17em}354144},\mathrm{\dots }$$

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, andso on.