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Problem 823 — Erdős Problems

Problem 823

Let $\alpha\geq 1$ . Is there a sequence of integers $n_k,m_k$ such that $n_k/m_k\to \alpha$ and $\sigma(n_k)=\sigma(m_k)$ for all $k\geq 1$ , where $\sigma$ is the sum of divisors function?

Source: erdosproblems.com/823

Number theory

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An accepted solution exists. The statement is true.

Proved. The settling result is recorded on the claim page Pollack 2015.

Er74b: Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202.
Po15b: Pollack, Paul, Remarks on fibers of the sum-of-divisors function. In: Analytic Number Theory, Springer, Cham (2015), 305-320, doi:10.1007/978-3-319-22240-0_18.

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Status: Solved, Proved
Topics: Number theory
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