There is no Carmichael number of the form \(2^n p^2+1\) with \(p\) prime (Q6546714)

An integer \(N\) is a Carmichael number provided it satisfies the congruence \(a^N\equiv a\pmod{N}\) for every integer \(a\); i.e., if they are Fermat pseudoprimes for every base \(a\).\N\NKorselt's criterion states that and integer \(N\) is a Carmichael number if and only if it is square-free and \(p-1\mid N-1\) for every prime \(p\mid N\).\N\NThe smallest Carmichael number is 561 and it is known since 1994 that there are infinitely many such numbers [\textit{W. R. Alford} et al., Ann. Math. (2) 139, No. 3, 703--722 (1994; Zbl 0816.11005)].\N\NIn this paper, it is proved that there is no Carmichael number of the form \(2^np^2+1\) with \(p\) prime. This builds upon previous work by one of the authors, in which it was proved that there is no Carmichael number of the form \(2^np+1\) with \(p\) prime [\textit{A. Alahmadi} and \textit{F. Luca}, C. R., Math., Acad. Sci. Paris 360, 1177--1181 (2022; Zbl 1504.11013)].