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Problem 229

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Four Representations using Squares

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 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 2×10⁹?

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四种平方数表示

3600是个特别的数，因为

3600 = 48² +      36² 3600 = 20² + 2×40² 3600 = 30² + 3×30² 3600 = 45² + 7×15²

相似地，我们发现88201 = 99² + 280² = 287² + 2×54² = 283² + 3×52² = 197² + 7×84²。

在1747年，欧拉证明了哪些数字可以表示成两个平方数之和。
我们感兴趣的则是那些数字n可以写成如下四种形式的平方数之和：

n = a₁² +    b₁² n = a₂² + 2 b₂² n = a₃² + 3 b₃² n = a₇² + 7 b₇²,

其中a_k和b_k均为正整数。

在不超过10⁷的数中，满足这一性质的有75373个。
在不超过2×10⁹的数中，满足这一性质的有多少个？

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