Carmichael numbers of the form $2^n p^m+1$ with odd exponent $m$: refinements of a theorem of Florian Luca, a general independence lemma, and a computational study

Abstract

Florian Luca devoted the last paper of his life to the set $\mathcal K$ of odd positive integers $k$ such that $2^n k+1$ is a Carmichael number for some positive integer $n$. He proved that if $m\ge 5$ is fixed and odd, then there are only finitely many primes $p$ with $p^m\in\mathcal K$. In this note, written as a small tribute to his memory, we extend his result in three directions, using only his published work as our reference base. First, we extract from his arguments explicit quantitative bounds: every Carmichael number $N=2^n p^m+1$ with $m$ odd satisfies $n<2^{2\cdot10^7(m+1)^2m^2(\log p)^2}$, and if $p>m$ the much sharper bound $n<25m^4\log p$ holds; for $p>500m^5\log m$ we further record the uniform bounds $ω(N)\le 26m^7$ on the number of prime factors. As an application, the number of such Carmichael numbers up to $x$ is $O_m(x^{1/m})$. Second, we generalize the multiplicative independence step of Luca's proof from base $2$ to an arbitrary even base $b$: if $b$ is even, $m\ge3$ is odd and coprime to $b$, $p>m$ is prime and $N=b^n p^m+1$ is Carmichael, then $b^n p^m$ and $b^a p^β$ are multiplicatively independent whenever $b^a p^β+1$ is a prime factor of $N$. Third, we report an exhaustive computation showing that the only Carmichael number $2^n p^m+1\le 10^{18}$ with odd $3\le m\le 37$ is $1729=2^6\cdot 3^3+1$, that no such number exists with odd $5\le m\le 37$ (extended to $10^{21}$ for $5\le m\le 15$), and that the only example with even $4\le m\le 36$ is $46657=2^6\cdot 3^6+1=13\cdot 37\cdot 97$, where $3^6=27^2$ is the square of the smallest element of $\mathcal K$.