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Updated Oct 7

Problem 830

Statement. We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$ . Are there infinitely many amicable pairs? If $A(x)$ counts the number of amicable $1\leq a\leq b\leq x$ then is it true that

A(x)>x^{1-o(1)}?

Status. Open.

Source. erdosproblems.com/830, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #830, https://www.erdosproblems.com/830.

References.

[Er55b] Erdős, P., On amicable numbers. Publ. Math. Debrecen (1955), 108-111.
[Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; doi:10.1007/978-0-387-26677-0. Section B4 "Amicable numbers" is on pp. 86--87; the conjecture $A(x)\geq x^{1-\epsilon}$ and the bounds $A(x)=o(x)$ and $A(x)\ll x\exp\{-(\ln x)^{1/3}\}$ are on p. 87. Library home: guy_2004_unsolved_problems_number_theory.
[Po15] Pomerance, Carl, On amicable numbers. Analytic Number Theory, Springer (2015), 321-327; doi:10.1007/978-3-319-22240-0_19.
[Po81] Pomerance, Carl, On the distribution of amicable numbers. II. J. Reine Angew. Math. 325 (1981), 183-188; doi:10.1515/crll.1981.325.183.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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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.

Linked from (14)

Arithmetic Functions Arithmetic Functions arithmetic_functions/erdos_1955_amicable_numbers Conjecture (p. 108): more than n^{1-ε} amicable numbers below n Lemma 1 (p. 109): almost every n is divisible by many primes from a sequence with divergent reciprocal sum Theorem (p. 110): the amicable numbers have density 0 arithmetic_functions/pollack_2016_problems_erdos_sum_divisors_function Theorem 1.1 (p. 2): up-down and down-up aliquot reversals in [1,x] number at most x/exp((sqrt 3+o(1))(log_3 x log_4 x)^{1/2})arithmetic_functions/pomerance_1981_distribution_amicable_numbers Theorem (p. 184): the amicable numbers up to x number at most x exp(-(log x)^{1/3}) for large x arithmetic_functions/pomerance_2015_amicable_numbers Lemma 2.1 (p. 3): squarefree n <= x with P(sigma(n)) <= y number at most x exp(-(1+o(1)) u log log u)Theorem 1.1 (p. 2): the amicable numbers up to x number at most x/exp((1/2+o(1)) sqrt(log x log log log x))number_theory/guy_2004_unsolved_problems_number_theory

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