Problem 824

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claims/: The 0 claim pages of Problem 824, one per claimant's result; the problem's standing derives from them.

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Statement. Let $h(x)$ count the number of integers $1\leq a<b<x$ such that $(a,b)=1$ and $\sigma(a)=\sigma(b)$, where $\sigma$ is the sum of divisors function.

Is it true that $h(x)>x^{2-o(1)}$?

Status. Open. The site labels the problem OPEN (page last edited 28 September 2025). Pollack and Pomerance [PoPo16] proved $h(x)>x^{1.4}$ for all large $x$, which gives $h(x)/x\to\infty$; Erdős [Er74b, p. 202] had sketched only $\limsup h(x)/x=\infty$, writing that the proof of $h(x)/x\to\infty$ "can be produced with a little more trouble". A partial result is claimed on the site's proof-claims tab (claim by Cam, using GPT 5.6 high, submitted 2026-08-28): the write-up claims $h(x)>x^{\gamma}$ for every fixed $\gamma<13/8$ and all large $x$, with squarefree pairs, by the scheme of [PoPo16] with Pascadi's level $5/8$ of distribution for primes in place of Baker and Harman's count of smooth shifted primes. A bound below exponent $2$ settles no instance of the question, so the claim has no claim page; the proof-claims thread had no comments as of 2026-10-06.

Source. erdosproblems.com/824, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #824, https://www.erdosproblems.com/824.

References.

• [Er74b] Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202.
• [PoPo16] Pollack, Paul and Pomerance, Carl, Some problems of Erdős on the sum-of-divisors function. Trans. Amer. Math. Soc. Ser. B (2016), 1-26.

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_2016_problems_erdos_sum_divisors_function
• pollack_2016_problems_erdos_sum_divisors_function / theorem_p24
• erdos_1974_remarks_problems_number_theory