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We give a new characterization of the set $\mathcal{C}$ of Carmichael numbersin the context of $p$-adic theory, independently of the classical results ofKorselt and Carmichael. The characterization originates from a surprising linkto the denominators of the Bernoulli polynomials via the sum-of-base-$p$-digitsfunction. More precisely, we show that such a denominator obeys atriple-product identity, where one factor is connected with a $p$-adicallydefined subset $\mathcal{S}$ of the squarefree integers that contains$\mathcal{C}$. This leads to the definition of a new subset $\mathcal{C}'$ of$\mathcal{C}$, called the "primary Carmichael numbers". Subsequently, weestablish that every Carmichael number equals an explicitly determinedpolygonal number. Finally, the set $\mathcal{S}$ is covered by modular subsets$\mathcal{S}_d$ ($d \geq 1$) that are related to the Kn\"odel numbers, where$\mathcal{C} = \mathcal{S}_1$ is a special case.

On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-$p$ digits