Kellner, Bernd C. On primary Carmichael numbers. (English) Zbl 1495.11008 Integers 22, Paper A38, 39 p. (2022). 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. Reviewer: Antonio M. Oller Marcén (Zaragoza) Cited in 1 Document MSC: 11A51 Factorization; primality 11A63 Radix representation; digital problems Keywords:Carmichael number; primary Carmichael number; taxycab number; polygonal number Citations:Zbl 1479.11028; Zbl 0020.34401 PDF BibTeX XML Cite \textit{B. C. Kellner}, Integers 22, Paper A38, 39 p. (2022; Zbl 1495.11008) Full Text: arXiv Link Online Encyclopedia of Integer Sequences: Carmichael numbers: composite numbers n such that a^(n-1) == 1 (mod n) for every a coprime to n. Taxicab, taxi-cab or Hardy-Ramanujan numbers: the smallest number that is the sum of 2 positive integral cubes in n ways. Carmichael numbers of the form (6*k+1)*(12*k+1)*(18*k+1), where 6*k+1, 12*k+1 and 18*k+1 are all primes. Cubefree taxicab numbers: the smallest cubefree number that is the sum of 2 positive cubes in n ways. 3-Carmichael numbers: Carmichael numbers equal to the product of 3 primes: k = p*q*r, where p < q < r are primes such that a^(k-1) == 1 (mod k) if a is prime to k. Least primary Carmichael number (A324316) with n prime factors. Primary Carmichael numbers. Number of primary Carmichael numbers (A324316) less than 10^n. Numbers m > 1 such that there exists a divisor g > 1 of m which satisfies s_g(m) >= g. Numbers m > 1 such that there exists a divisor g > 1 of m which satisfies s_g(m) = g. Numbers m > 1 such that every prime divisor p of m satisfies s_p(m) >= p. Numbers m > 1 such that every prime divisor p of m satisfies s_p(m) = p. 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