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Statement

Setting (p. 108). $\sigma(n)$ is the sum of all divisors of $n$, $n$ included, and two numbers $a,b$ are amicable when $\sigma(a)=\sigma(b)=a+b$; equivalently, with $\sigma_1(a)=\sigma(a)-a$ the sum of the divisors of $a$ below $a$, when $\sigma_1(a)=b$ and $\sigma_1(b)=a$. An amicable number is a member of such a pair.

Theorem (p. 110, unnumbered, quoted). "The density of amicable numbers is 0."

That is, the number of amicable numbers up to $x$ is $o(x)$. The paper sets it against the recent bound of Kanold (p. 108), that the density of the amicable numbers is less than $0{\cdot}204$.

Statements without proof (p. 108). The paper asserts that its method could show that fewer than $c\cdot n/\log\log\log n$ amicable numbers lie below $n$, for some constant $c$; no argument for this bound is written out. It adds that the count below $n$ is no doubt $o\bigl(n/(\log n)^k\bigr)$ for every $k$, which the method does not seem able to reach. Nor is the assertion (pp. 108--109) argued that the method would show, for every $k$, that the integers $a$ with $\sigma_1^{(k)}(a)=a$ have density 0, where $\sigma_1^{(k)}$ is the $k$-fold iterate of $\sigma_1$.

Source. P. Erdős, On amicable numbers, Publ. Math. Debrecen 4 (1955), 108--111: the theorem stated on p. 110, its proof on pp. 110--111. The edition read is identified on the source card.

Read depth. Claims checked: the statement and the setting were read clause by clause on the printed pages. The proof was followed for structure and not verified. Nothing here is independently reviewed.

Proof pointer

Pp. 110--111. List the amicable pairs as $(a_i,b_i)$ with $a_i<b_i$; it suffices to show that the $a_i$ have density 0. Fix a large $A$. The $a_i$ for which $\sigma(a_i)$ fails to be divisible by $p^A$ for some prime $p\le A$ have density 0 by Lemma 2. For the others, every $d\le A$ dividing $a_i$ divides $\sigma(a_i)$ and hence divides $b_i=\sigma(a_i)-a_i$. By Lemma 3 the divisors up to $A$ account for all but $\eta a_i$ of $\sigma(a_i)$ outside a set of at most $\varepsilon x$ of the $a_i$ up to $x$, and as they also divide $b_i$ this gives $\sigma(b_i)/b_i\ge\sigma(a_i)/a_i-\eta$. With $\sigma(a_i)=\sigma(b_i)=a_i+b_i$ this forces $1<b_i/a_i<1+\eta$, hence $2<\sigma(a_i)/a_i<2+\eta$. The density of the integers with $\sigma(n)/n\le c$ exists and is a continuous function of $c$ (the paper cites Davenport, 1933, and two papers of Erdős, 1934 and 1935), so for $\eta$ small these $a_i$ have density below $\varepsilon$.

Dependencies

• Lemma 1 of the same paper, through Lemma 2.
• Lemma 2 (p. 109): for every constant $A$, the integers $n$ for which $\sigma(n)$ is not divisible by $\bigl(\prod_{p\le A}p\bigr)^A$ have density 0. The proof applies Lemma 1 to the primes $q\equiv-1\pmod p$ that exceed an arbitrary bound $B$.
• Lemma 3 (p. 110): with $\sigma_A(n)=\sum_{d\mid n,\,d\le A}n/d$, for every $\varepsilon$ and $\eta$ there is an $A_0$ such that for $A>A_0$ fewer than $\varepsilon x$ integers $n\le x$ have $\sigma(n)-\sigma_A(n)>\eta n$. The proof is a first-moment count.
• The continuity of the distribution function of $\sigma(n)/n$, cited from Davenport (Sitzungsber. Preuß. Akad. Wiss. 1933) and from Erdős (J. London Math. Soc. 9 (1934) and 10 (1935)).

Bears on

• Problem 830: the problem's $A(x)$ counts amicable pairs with both members at most $x$, which is at most the number of amicable numbers up to $x$, so the theorem gives the upper bound $A(x)=o(x)$. It gives no lower bound and does not decide whether there are infinitely many amicable pairs.