There are no Carmichael numbers of the form \(2^np+1\) with \(p\) prime (Q2080964)

A Sierpinski number is an odd positive integer \(k\) such that \(2^nk+1\) is composite for every positive integer \(n\). A Carmichael number is an odd integer \(N\) such that it is composite but \(a^N\equiv a\pmod{N}\) for every \(a\). This paper relates both concepts by considering the set \[ \mathcal{K}=\{k \text{ odd}: \{2^nk+1\}_{n\geq 0} \text{ contains some Carmichael number}\} \] In particular it is proved that all members of \(\mathcal{K}\) are in fact composite. Note that this is equivalent to saying that there is no Carmichael number of the form \(2^np+1\) with \(p\) an odd prime.