The n-conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a positive integer $m$, let $\operatorname{rad}(m)=\prod_{p\mid m}p$, where the product is over distinct prime divisors and $\operatorname{rad}(1)=1$. An integer tuple $(a_1,\ldots,a_n)$ is collectively coprime when $\gcd(|a_1|,\ldots,|a_n|)=1$. It has no vanishing proper subsum when $\sum_{i\in I}a_i\ne0$ for every nonempty proper subset $I\subsetneq\{1,\ldots,n\}$.

Statement recorded in the supplied source.

For every integer $n\ge3$ and every real $\varepsilon>0$, does there exist a constant $C_{n,\varepsilon}>0$ such that every collectively coprime tuple of nonzero integers with \[\sum_{i=1}^n a_i=0\] and no vanishing proper subsum satisfies \[\max_{1\le i\le n}|a_i|<C_{n,\varepsilon}\operatorname{rad}(|a_1\cdots a_n|)^{2n-5+\varepsilon}?\] [S2]

Short English statement

For each tuple length, is the largest absolute entry uniformly bounded by the stated power of the distinct primes dividing the product, provided the entries are collectively coprime and no smaller subsum vanishes?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400088 (AMR-093-0088). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, The n-conjecture. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.
• S3 Wikipedia contributors, N conjecture, revision 1378050537 (2026) Statement, first formulation. https://en.wikipedia.org/w/index.php?title=N+conjecture&oldid=1378050537.