Selfridge-Wagstaff-Pomerance Prize Problem

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 $(F_j)_{j\geq0}$ be the Fibonacci sequence with $F_0=0$, $F_1=1$ and $F_{j+2}=F_{j+1}+F_j$. A composite number is an integer greater than one that is not prime. Congruence $n\equiv3\text{ or }7\pmod{10}$ makes $n$ odd and coprime to $10$.

Statement recorded in the supplied source.

Does there exist a composite integer $n$ satisfying \[n\equiv3\text{ or }7\pmod{10},\qquad n\mid(2^n-2),\qquad n\mid F_{n+1}?\] Find such an integer or prove that none exists. [S2]

Short English statement

Can a composite integer ending in 3 or 7 simultaneously pass the stated base-two divisibility test and divide the next Fibonacci number?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1876 (GUY-A12b). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Andrew Shallue and Jonathan Webster, Fast tabulation of challenge pseudoprimes, arXiv:1806.08697 (2018); Open Book Series 2 (2019), 411--428. Introduction and Section2, pages1--3. https://arxiv.org/pdf/1806.08697.