Problem 823

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claims/: The 1 claim page of Problem 823, one per claimant's result; the problem's standing derives from them.

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Statement. 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?

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

Source. erdosproblems.com/823, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #823, https://www.erdosproblems.com/823.

References.

• [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.

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• pollack_2015_remarks_fibers_sum_divisors_function
• pollack_2015_remarks_fibers_sum_divisors_function / theorem_1
• erdos_1974_remarks_problems_number_theory