# Problem 229: Four Representations using Squares

![graphic](img229.gif)

Consider the number 3600. It is very special, because 3600 = 482 +   
 362 3600 = 202 + 2×402 3600 = 302 + 3×302 3600 = 452 + 7×152 Similarly,
we find that 88201 = 992 + 2802 = 2872 + 2×542 = 2832 + 3×522 = 1972 +
7×842. In 1747, Euler proved which numbers are representable as a sum of
two squares. We are interested in the numbers n which admit
representations of all of the following four types: n = a12 +   b12n =
a22 + 2 b22n = a32 + 3 b32n = a72 + 7 b72, where the ak and bk are
positive integers. There are 75373 such numbers that do not exceed 107.
How many such numbers are there that do not exceed 2×109?