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I am trying to find a conjecture apparently made by Erdős and Straus. I say apparently because I have had so much trouble finding anything information about it that I'm beginning to doubt its existence. Here it is:

Let $\phi(X)$ be a rational function over $\mathbb{Q}$ that is defined at every positive integer. If the sum $\sum \limits_{n=1}^\infty \phi(n)$ converges, it is either rational or transcendental, i.e., it is never an irrational algebraic number.

Has anyone heard of this conjecture? I was told about it by my supervisor, but he doesn't remember where he heard about it.

  • 8
    I was talking with a friend today and he mentioned that "On a Conjecture of Erdos" is the most popular paper title.2011-12-02
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    One can always perform a partial fraction decomposition of $\phi(X)$, and then be able to sum in terms of either polygamma functions or (generalized) harmonic numbers.2011-12-02
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    Further hypothesis needed, else $\phi(X)=\sqrt2/(\pi^2X^2)$ is a counterexample.2011-12-02
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    [This](http://www.sku.ac.ir/academic/members/hessami-k/files/nvsumsfinal.pdf) is quite relevant...2011-12-02
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    I assume $\phi$ is supposed to have rational coefficients.2011-12-02
  • 0
    Oops. Yes, $\phi$ should have rational coefficients.2011-12-06

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