Carmichael numbers in the sequence (k2^n+1)_{n\ge 1}

TL;DR

This paper proves that for each odd k, the sequence (k2^n+1) contains finitely many Carmichael numbers, and identifies 27 as the smallest k with such a Carmichael number.

Contribution

It establishes finiteness of Carmichael numbers in the sequence for odd k and determines the minimal k value with such numbers.

Findings

Finiteness of Carmichael numbers in (k2^n+1) for odd k

27 is the smallest k with a Carmichael number in the sequence

Provides new insights into the distribution of Carmichael numbers

Abstract

We prove that for each odd number k, the sequence (k2^n+1)_{n\ge 1} contains only a finite number of Carmichael numbers. We also prove that k=27 is the smallest value for which such a sequence contains some Carmichael number.