Erdos 1955 amicable numbers

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conjecture_p108: Erdős's conjecture that for every eps > 0 the number of amicable numbers less than n exceeds n to the power 1 - eps once n > n_0(eps), with his remark that it is not known whether there are infinitely many.

lemma_1: For a sequence of primes q_i whose reciprocals have divergent sum, the integers n divisible by fewer than A of the q_i have density 0, for every A; Erdős derives it as a special case of a theorem of Turán.

theorem_p110: Erdős's theorem that the set of amicable numbers, the a for which some b satisfies sigma(a) = sigma(b) = a + b, has asymptotic density 0.

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P. Erdős: On amicable numbers, Publ. Math. Debrecen 4 (1955), 108--111; MR 16,998h; Zentralblatt 65,27. The copy read for this card is the Rényi Institute's Erdős archive scan, which prints no notice; the journal's site prints the footer "© 2026, Publicationes Mathematicae, Debrecen, Hungary" and offers volumes 1–95 as a free archive naming no license (https://publi.math.unideb.hu/, read 2026-10-02), every other right reserved.

Erdős proves that the set of amicable numbers (a with sigma(a) = sigma(b) = a + b for some b) has density 0, superseding Kanold's bound of density below 0.204. Erdős states without proof that the method could show that fewer than c n / log log log n amicable numbers lie below n, and he remarks that the true count is no doubt o(n/(log n)^k) for every k, which the method does not seem able to reach; he also conjectures that more than n^{1-eps} amicable numbers lie below n for every eps > 0 and n > n_0(eps). The proof rests on several lemmas, the first being that for a sequence of primes q_i with divergent sum of reciprocals the density of n divisible by fewer than A of the q_i is 0 for every A, a special case of a theorem of Turán on additive functions. Erdős also remarks, without proof, that it would not be hard to prove by the method that for every k the density of a with sigma_1^{(k)}(a) = a is 0, where sigma_1(a) = sigma(a) - a, while Catalan's conjecture that the iterated sequence sigma_1^{(n)}(a) is bounded, and whether the density of a with sigma_1^{(n)}(a) = 1 for some n exists, seem inaccessible. A closing remark (p. 111) states that Lemmas 2 and 3 give, for every eps, density 0 for the a with sigma(b)/b < sigma(a)/a - eps where b = sigma(a) - a, and that a more complicated argument, not given, gives the same for sigma(b)/b > sigma(a)/a + eps, so that sigma(b)/b = sigma(a)/a + o(1) outside a set of density 0.

Source: https://users.renyi.hu/~p_erdos/1955-03.pdf.

Results

Labels and pages are those of the journal print, pp. 108--111.

• Conjecture (p. 108): for every $\varepsilon>0$, more than $n^{1-\varepsilon}$ amicable numbers lie below $n$ once $n>n_0(\varepsilon)$; the paper notes that it is not known whether there are infinitely many.
• Lemma 1 (p. 109): if primes $q_i$ satisfy $\sum_i1/q_i=\infty$, then for every $A$ the integers divisible by fewer than $A$ of the $q_i$ have density 0, a special case of a theorem of Turán.
• Theorem (p. 110; proof pp. 110--111): the amicable numbers have density 0. The page also records the bound $c\,n/\log\log\log n$ and the remark on the iterates of $\sigma_1$ that the paper states without proof.

Lemma 2 (p. 109) and Lemma 3 (p. 110) are proof steps, stated in the theorem's page under its dependencies.

Bears on.

• #830: the Theorem gives the upper bound $A(x)=o(x)$, since $A(x)$ is at most the number of amicable numbers up to $x$; the paper does not decide whether there are infinitely many amicable pairs. The Conjecture has the shape of the problem's lower bound but counts amicable numbers below $n$ rather than pairs with both members at most $x$; the problem's bound implies it, and the paper does not compare the two counts in the other direction.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.