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Four Representations using Squares
Problem 229
Published on 24 January 2009 at 09:00 am [Server Time]
Consider the number 3600. It is very special, because
3600 = 482 + 362
3600 = 202 + 2×402
3600 = 302 + 3×302
3600 = 452 + 7×152
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 + 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 2×109?
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