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 $p \le 1000$, 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$p \le 1000$中，$p$取何值时有最多个解？