A Solution of Sierpinski Problem Based m

TL;DR

This paper generalizes Sierpinski's problem by proving that for any integer greater than 1, there are infinitely many numbers k such that km^n+1 is composite for all n, using covering systems and cyclotomic polynomials.

Contribution

It introduces a new generalization of Sierpinski numbers for arbitrary bases m, extending the original problem and providing a proof using advanced number theory techniques.

Findings

Infinitely many Sierpinski numbers based m exist for any m>1.

Theorem applies to all positive integers m>1.

Uses covering systems and cyclotomic polynomials in the proof.

Abstract

In 1960, W. Sierpinski proved that there are infinitely many positive odd numbers $k$, such that for any positive integer $n$, $k\times2^n+1$ is a composite number. Such numbers are called "Sierpinski numbers". In this study, by using covering systems and the theory of cyclotomic polynomials, the following theorem is proved: for any integer $m>1$, there are infinitely many integers $k$ satisfying $k\not\equiv-1\!\pmod{q}$ for any prime number $q|(m-1)$, such that for any positive integer $n$, $km^n+1$ is a composite number. These positive integers $k$ are called "Sierpinski numbers based $m$". The theorem can be regarded as a generalization of Sierpinski problem.