Problem 39

Integer Right Triangles

If $p$ is the perimeter of a right angle triangle with integral length sides, ${a, b, c}$ , there are exactly three solutions for $p = 120$ .

${20, 48, 52}, {24, 45, 51}, {30, 40, 50}$

For which value of $p \leq 1000$ , is the number of solutions maximised?

考虑三边长 ${a, b, c}$ 均为整数的直角三角形，并记其周长为 $p$ ，当 $p = 120$ 时，恰好存在三个不同的解：

${20, 48, 52}, {24, 45, 51}, {30, 40, 50}$

在所有的 $p \leq 1000$ 中， $p$ 取何值时有最多个解？