Problem 229
Four Representations using Squares
Consider the number 3600. It is very special, because
3600 = 482 + 362
3600 = 202 + 2
402
3600 = 302 + 3302
3600 = 452 + 7152
3600 = 202 + 2
4023600 = 302 + 3
3023600 = 452 + 7
152Similarly, 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,
n = a22 + 2 b22
n = a32 + 3 b32
n = 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 2109?
These problems are part of Project Euler and are licensed under CC BY-NC-SA 2.0 UK