6.2 Primitive Pythagorean triangles with square perimeters 211
6.2 Primitive Pythagorean triangles with square perime-
ters
Ifm>nare relatively prime integers of opposite parity, they generate a
primitive Pythagorean triple(a, b, c)=(m^2 −n^2 , 2 mn, m^2 +n^2 )with
perimeterp=2m(m+n). If this perimeter is a square (number), we
must havem=2q^2 andm+n=p^2 for some integerspandq. From
these,(m, n)=(2q^2 ,p^2 − 2 q^2 ).
a=m^2 −n^2 =p^2 (4q^2 −p^2 ),
b=2mn=4q^2 (p^2 − 2 q^2 ),
c=m^2 +n^2 =p^4 − 4 p^2 q^2 +8q^4.Note thatpis odd,√
2 q≤p< 2 q, andgcd(p, q)=1. The perime-
ter is 4 p^2 q^2 =(2pq)^2.
Here are the first few of such triangles. The last column gives the
square root of the perimeter.
pqmn abc 2 pq
328163 16 65 12
5318 7 275 252 373 30
7432 17 735 1088 1313 56
9550 31 1539 3100 3461 90
11 6 72 49 2783 7056 7585 132
11 7 98 23 9075 4508 10133 154
13 7 98 71 4563 13916 14645 182
13 8 128 41 14703 10496 18065 208
15 8 128 97 6975 24832 25793 240
13 9 162 7 26195 2268 26293 234
17 9 162 127 10115 41148 42373 306
17 10 200 89 32079 35600 47921 340
19 10 200 161 14079 64400 65921 380
..
.