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Four Representations using Squares

Problem 229

Published on Saturday, 24th January 2009, 09:00 am; Solved by 893

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 +   b12

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?