For a finite set A e textstylevarnothing of positive integers, let Sigma^(A) denote the set of nonempty subset sums, and let P(A) be the greatest prime factor of ∏_a∈ Aa. Erdős–Sárközy–Stewart's conjecture. For A⊆N, lim_|A|rightarrow∞fracP(Sigma^(A))|A|=∞. This conjecture asserts that the greatest prime factor appearing among the nonempty subset sums eventually dominates the size of the set, and is part of the paper's discussion of prime factors of subset sums.

Erdős–Sárközy–Stewart conjecture on prime factors of subset sums

For a finite set $A e\textstyle\varnothing$ of positive integers, let $\Sigma^*(A)$ denote the set of nonempty \subset sums, and let $P(A)$ be the greatest \prime factor of $\prod_{a\in A}a$ . Erdős–Sárközy–Stewart's conjecture. For $A\subseteq\mathbb{N}$ ,

\lim_{|A|\rightarrow\infty}\frac{P(\Sigma^*(A))}{|A|}=\infty.

This conjecture asserts that the greatest prime factor appearing among the nonempty subset sums eventually dominates the size of the set, and is part of the paper's discussion of prime factors of subset sums.

Sources & referencesView supporting material

Primary source

Ernie Croot, Junzhe Mao and Chi Hoi Yip, “Hilbert cubes in sets with arithmetic properties”, arXiv:2603.14654 (2026).

additive-combinatorics prime-factors

1 week ago

Progress summary

Refreshed 10 hours ago

Partially solved

The conjecture remains unproved, but a March 2026 paper obtained conditional partial consequences in a related smooth-number setting.

The conjecture asks whether the largest prime factor among the nonempty subset sums of a finite set eventually grows faster than the set's size. A recent paper records this as Conjecture $2.9$ and also states the stronger quadratic lower-bound conjecture.

March 2026 partial progress

“Hilbert cubes in sets with arithmetic properties” partially justifies consequences corresponding to both conjectures, but only under assumptions on the growth of a smoothness parameter. Its corollaries concern sets whose shifted subset sums are smooth numbers; they do not prove either the original conjecture or its stronger quadratic form.

Current status (as of August 2026): The Erdős–Sárközy–Stewart conjecture remains open, with only conditional partial progress in a related smooth-number setting; the stronger quadratic conjecture is also open.

Sources

arxiv.org

No solutions have been posted yet.