## On primary Carmichael numbers.(English)Zbl 1495.11008

Given an integer $$m$$ and a prime $$p$$, let us denote by $$s_p(m)$$ the sum of the base $$p$$ digits of $$m$$. If we denote by $$\mathbb{S}$$ he set of square-free integers greater than 1, then the set of primary Carmichael numbers is given by $\mathcal{C}'=\{m\in\mathbb{S}:p\mid m\Rightarrow s_p(m)=p\}.$ It was proved by B. C. Kellner and J. Sondow [Integers 21, Paper A52, 21 p. (2021; Zbl 1479.11028)] that the elements of $$\mathcal{C}'$$ are in fact Carmichael numbers; i.e., they satisfy Fermat’s little Theorem for every coprime base.
J. Chernick [Bull. Am. Math. Soc. 45, 269–274 (1939; Zbl 0020.34401)] provided a method to construct Carmichael numbers from certain polynomials under some mild assumptions. For example, $$U_3(t)=(6t+1)(12t+1)(18t+1)$$ is a Carmichael number whenever its three factors are odd primes.
Among other results, in the paper it is proved that in many cases $$U_3(t)$$ is not only a Carmichael number, but also a primary Carmichael number. Then, some connections to taxicab and polygonal numbers are investigated and exemplified.

### MSC:

 11A51 Factorization; primality 11A63 Radix representation; digital problems

### Citations:

Zbl 1479.11028; Zbl 0020.34401
Full Text:

### References:

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