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.


11A51 Factorization; primality
11A63 Radix representation; digital problems
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.
Numbers m > 1 that have an s-decomposition.
Numbers m > 1 that have a strict s-decomposition.
Numbers m > 1 such that there exists a prime divisor p of m with s_p(m) = p.
Numbers m > 1 such that there exists a composite divisor c of m with s_c(m) = c.
Squarefree polygonal numbers P(s,n) with s >= 3 and n >= 3.
Special polygonal numbers.
Rank of the n-th special polygonal number A324973(n).
Rank of the n-th Carmichael number.
Rank of the n-th primary Carmichael number.


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