Project Euler/229
From charlesreid1
Problem Statement
Four Representations Using Squares
Consider the number 3600. It is very special, because
3600 = 48^2 + 36^2 3600 = 20^2 + 2×40^2 3600 = 30^2 + 3×30^2 3600 = 45^2 + 7×15^2
Similarly, we find that 88201 = 99^2 + 280^2 = 287^2 + 2×54^2 = 283^2 + 3×52^2 = 197^2 + 7×84^2.
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 = a_1^2 + b_1^2 n = a_2^2 + 2 b_2^2 n = a_3^2 + 3 b_3^2 n = a_7^2 + 7 b_7^2,
where the a_k and b_k are positive integers.
There are 75373 such numbers that do not exceed 10^7.
How many such numbers are there that do not exceed 2×10^9?
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