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On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-\(p\) digits. (English) Zbl 1479.11028

A Carmichael number is a composite number \(m\) such that the congruence \(a^m\equiv 1 \pmod{m}\) holds for all integers \(a\) coprime to \(m\). This congruence holds for all prime \(m\) by Fermat’s little theorem. This paper complements two different characterizations of Carmichael numbers, due to Korselt and Carmichael. It draws new and interesting connections between Carmichael numbers and the function \(s_p(m)\), where \(s_p(m)\) denotes the sum of the digits of \(m\) expressed in base \(p\).
This paper defines {primary Carmichael numbers}, a subset of the Carmichael numbers given by those squarefree \(m\) such that \(s_p(m)= p\) for all prime divisors of \(m\). It also defines a subset of the integers containing the Carmichael numbers, given by those squarefree \(m\) such that \(s_p(m)\geq p\) for all prime divisors of \(m\).
This paper then offers a new characterization of the Carmichael numbers in terms of the sum of digits function, as squarefree \(m\) satisfying \[\{m: p|m \text{ implies } s_p(m)\geq p,\quad s_p(m) \equiv 1 \pmod{p-1}\},\] complementing the characterizations of Korselt and Carmichael. It then proves some results expressing the denominators of the Bernoulli polynomials in terms of \(s_p(m)\). It also provides another explicit characterization of the Carmichael numbers in terms of polygonal numbers and \(s_p(m)\). The proofs follow from from clever elementary number theoretic considerations.

MSC:

11A51 Factorization; primality
11B68 Bernoulli and Euler numbers and polynomials
11A63 Radix representation; digital problems
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