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On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-\(p\) digits. (English) Zbl 07354559
Summary: We give a new characterization of the set \(\mathcal{C}\) of Carmichael numbers in the context of \(p\)-adic theory, independently of the classical results of Korselt and Carmichael. The characterization originates from a surprising link to the denominators of the Bernoulli polynomials via the sum-of-base-\(p\)-digits function. More precisely, we show that such a denominator obeys a triple-product identity, where one factor is connected with a \(p\)-adically defined subset \(\mathcal{S}\) of the squarefree integers that contains \(\mathcal{C}\). This leads to the definition of a new subset \(\mathcal{C}^{\prime}\) of \(\mathcal{C}\), called the “primary Carmichael numbers”. Subsequently, we establish that every Carmichael number equals an explicitly determined polygonal number. Finally, the set \(\mathcal{S}\) is covered by modular subsets \(\mathcal{S}_d\) \((d \ge 1)\) that are related to the Knödel numbers, where \(\mathcal{C}=\mathcal{S}_1\) is a special case.
MSC:
11A51 Factorization; primality
11B68 Bernoulli and Euler numbers and polynomials
11A63 Radix representation; digital problems
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