Problem 229

Four Representations using Squares

Published on Saturday, 24th January 2009, 09:00 am; Solved by 826

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 + 254² = 283² + 352² = 197² + 784².

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 + b₁²

n = a₂² + 2 b₂²

n = a₃² + 3 b₃²

n = a₇² + 7 b₇²,

where the a_k and b_k are positive integers.

There are 75373 such numbers that do not exceed 10⁷.
How many such numbers are there that do not exceed 210⁹?