# Which numbers are sums of finite numbers of reciprocal squares?

Question: Is there a “nice” characterization of rational numbers $$q$$ for which $$q$$ can be written as

$$q = \frac{1}{n_1^2} + \frac{1}{n_2^2} + \dots + \frac{1}{n_k^2}$$

for distinct natural numbers $$n_1 \lt n_2 \lt \dots \lt n_k$$?

I’m aware of several related results, but none of them seem applicable here:

• Lagrange’s four-square theorem says that every natural number is the sum of four squares, but that doesn’t transfer to reciprocals of squares.
• The Basel problem states that $$\sum_{n=1}^\infty{1 \over n^2} = {\pi^2 \over 6}$$, which gives an upper bound on the numbers that can be written this way. If we allow for infinitely many terms in the sum I think (?) this would mean we can express every number up to $$\pi^2 \over 6$$ this way, but I’m interested only in finite sums.
• The existence of Egyptian fraction representations and the divergence of the harmonic series ensures every nonnegative rational number can be written as a sum of distinct unit fractions, but those fractions aren’t necessarily constrained to be inverse squares.

Is there a simple characterization of the rational numbers that can be written this way?

• Commented Jun 29 at 7:45

You can't express every rational number up to $$\pi^2/6$$. For example, $$3/4$$ is impossible - the sum can't contain 1, but then it's bounded above by $$\pi^2/6 - 1$$, which is less than $$3/4$$. However this sort of argument only seems to exclude the interval $$[\pi^2/6 - 1, 1)$$.
And it turns out (much less obviously) that that's the only obstruction - see R. L. Graham, On Finite Sums of Reciprocals of Distinct nth powers, Pacific Journal of Mathematics, 1964. In particular Corollary 1 says a rational $$p/q$$ is a sum of distinct reciprocals of squares if and only if $$p/q \in [0, \pi^2/6-1) \cup [1, \pi^2/6).$$
• I meant $[\pi^2/6 - 1, 1)$ - fixed. Commented Jun 28 at 16:15