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Statement

Conjecture (p. 609, unnumbered, quoted). "The order of the maximum $n(k)$ of $n$ belonging to a given number $k$ of primes is probably $n(k)=O(k^{1+\epsilon})$ for any $\epsilon>0$ but actually we cannot prove this relation."

Here $n$ is the number of positive integers $a_1,\ldots,a_n$ whose two-term sums $a_i+a_j$, $i\ne j$, contain no prime factor other than the $k$ given primes (p. 608). Footnote 1 (p. 609) defines $f(x)=O(g(x))$ as the existence of $B$ and $A$ with $\lvert f(x)\rvert<Ag(x)$ for all $x\ge B$. The sentence before the conjecture says that the bound of Theorem I, $n(k)\le3\cdot2^{k-1}-1$, is probably not exact.

Source. Paul Erdős and Paul Turán, On a problem in the elementary theory of numbers, Amer. Math. Monthly 41 (1934), 608-611: the conjecture and its footnote on p. 609. The edition read is identified on the source card.

Read depth. Claims checked: the sentence and its footnote were read on the printed page. The paper gives no argument for it.

Proof pointer

None; the paper states that it cannot prove the relation.

Dependencies

None.

Bears on

• Problem 126: for a set of $n$ distinct positive integers whose product of pairwise sums has $k$ distinct prime factors, $n\le n(k)$. The conjecture would therefore give, for every $\epsilon>0$, at least $c_\epsilon n^{1/(1+\epsilon)}$ distinct prime factors for some $c_\epsilon>0$, and with it $f(n)/\log n\to\infty$ (an observation of this page, not of the paper). The square-root bound recorded on the Adamczewski 2026 result page corresponds to $n(k)=O(k^2)$, which does not reach the conjectured order.