Selfridge's 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.

A Sierpiński number is an odd positive integer $k$ for which $k2^n+1$ is composite for every integer $n\ge1$. Here composite means an integer greater than one which is not prime. The integer $78557$ is a known such number.

Statement recorded in the supplied source.

Is $78557$ the least Sierpiński number? Equivalently, does every odd integer $k$ with $1\le k<78557$ have some $n\ge1$ for which $k2^n+1$ is prime? [S2]

Short English statement

Is seventy-eight thousand five hundred fifty-seven the smallest odd multiplier whose entire exponential sequence plus one is composite?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400136 (AMR-093-0136). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Selfridge's conjecture. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.
• S3 OEIS Foundation and contributors, The On-Line Encyclopedia of Integer Sequences, A076336: (Provable) Sierpiński numbers: odd numbers n such that for all k >= 1 the numbers n*2^k + 1 are composite., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A076/A076336.seq.