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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