Math-4194 · mathematics · Number theory

Fortune's conjecture

Status Open · unsolvedmath.comLLM status Open unverified

Statement

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.

Let $p_1=2<p_2=3<\cdots$ be the primes and let $P_n=\prod_{j=1}^{n}p_j$ for $n\ge1$. Define the $n$th Fortunate number by \[ F_n=\min\{m\in\mathbb N:m>1,\ P_n+m\text{ is prime}\}. \] The set is nonempty because primes are unbounded. The condition $m>1$ is part of the definition.

Statement recorded in the supplied source.

Is $F_n$ prime for every $n\ge1$? [S2]

Short English statement

Is the smallest increment greater than one which turns each primorial into a prime always itself prime?

Sources

S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400124 (AMR-093-0124). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
S2 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Fortune'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, A005235: Fortunate numbers: least m > 1 such that m + prime(n)# is prime, where p# denotes the product of all primes <= p., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A005/A005235.seq.

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

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Record details

Subjects: Number theory
Collection: Top Open Problems
Mathematics rank: 4194
UnsolvedMath: Catalogue entry