Wikipedia number-theory item 174: For any given integers $k\geq 1, b\geq 2, c\neq 0$, with gcd(k, c) = 1 and gcd(b, c) = 1, are there…

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.

Fix integers $k\ge1$, $b\ge2$ and $c\ne0$ with $\gcd(k,c)=\gcd(b,c)=1$, where greatest common divisors are positive. Put $d=\gcd(k+c,b-1)$ and $A_n=(kb^n+c)/d$ for $n\ge1$. Since $b\equiv1\pmod d$, every $A_n$ is an integer. A prime value means $A_n>1$ is prime. As written, the universal assertion has a counterexample: $k=78557$, $b=2$, $c=1$ meets all the hypotheses and has $d=1$, while every $A_n=78557\cdot2^n+1$ is composite because $78557$ is a Sierpiński number. [S3]

Statement recorded in the supplied source.

For every triple $(k,b,c)$ satisfying the displayed hypotheses, are there infinitely many $n\ge1$ for which $A_n$ is prime? The stated hypotheses alone do not imply this conclusion, as the displayed Sierpiński counterexample shows. [S2]

Short English statement

Does every indicated exponential sequence have infinitely many prime values? The stated conditions allow a known sequence consisting entirely of composite values.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400174 (AMR-093-0174). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Wikipedia number-theory item 174: For any given integers $k\geq 1, b\geq 2, c\neq 0$, with gcd(k, c) = 1 and gcd(b, c) = 1, are there…. 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.