Rational Number Expressible as Sum of Reciprocals of Distinct Squares

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Theorem

Let $x$ be a rational number such that $0 < x < \dfrac {\pi^2} 6 - 1$.

Then $x$ can be expressed as the sum of a finite number of reciprocals of distinct squares.


Proof

That no rational number such that $x \ge \dfrac {\pi^2} 6 - 1$ can be so expressed follows from Riemann Zeta Function of 2:

$\ds \sum_{n \mathop = 1}^n \dfrac 1 {n^2} = 1 + \dfrac 1 {2^2} + \dfrac 1 {3^2} + \dotsb = \dfrac {\pi^2} 6$

That is, using all the reciprocals of distinct squares, you can never get as high as $\dfrac {\pi^2} 6 - 1$.

It remains to be shown that for all rational numbers $x$ less than $\dfrac {\pi^2} 6 - 1$, you can make $x$ with a subset of them.


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Examples

Example: $\dfrac 1 2$

$\dfrac 1 2$ can be expressed as the sum of a finite number of reciprocals of distinct squares as follows:

$\dfrac 1 2 = \dfrac 1 {2^2} + \dfrac 1 {3^2} + \dfrac 1 {4^2} + \dfrac 1 {5^2} + \dfrac 1 {7^2} + \dfrac 1 {12^2} + \dfrac 1 {15^2} + \dfrac 1 {20^2} + \dfrac 1 {28^2} + \dfrac 1 {35^2}$


Sources

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