##
**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.

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)

### 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.

### References:

[1] | W. R. Alford, A. Granville, and C. Pomerance, There are infinitely many Carmichael numbers, Ann. of Math. 139 (1994), 703-722. · Zbl 0816.11005 |

[2] | C. S. Calude, E. Calude, and M. J. Dinneen, What is the value of Taxicab(6)?, J. Universal Computer Science 9 (2003), 1196-1203. |

[3] | R. D. Carmichael, Note on a new number theory function, Bull. Amer. Math. Soc. 16 (1910), 232-238. · JFM 41.0226.04 |

[4] | R. D. Carmichael, On composite numbers P which satisfy the Fermat congruence a P −1 ≡ 1 (mod P ), Amer. Math. Monthly 19 (1912), 22-27. · JFM 42.0236.07 |

[5] | J. Chernick, On Fermat’s simple theorem, Bull. Amer. Math. Soc. 45 (1939), 269-274. · JFM 65.0123.02 |

[6] | H. Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM 240, Springer-Verlag, New York, 2007. |

[7] | J. H. Conway and R. K. Guy, The Book of Numbers, Springer-Verlag, New York, 1996. · Zbl 0866.00001 |

[8] | L. E. Dickson, A new extension of Dirichlet’s theorem on prime numbers, Messenger 33 (1904), 155-161. |

[9] | H. Dubner, Carmichael numbers of the form (6m + 1)(12m + 1)(18m + 1), J. Integer Seq. 5 (2002), Article 02.2.1, 1-8. · Zbl 1020.11005 |

[10] | A. Granville and C. Pomerance, Two contradictory conjectures concerning Carmichael num-bers, Math. Comp. 71 (2002), 883-908. · Zbl 0991.11067 |

[11] | U. Hollerbach, The sixth taxicab number is 24 153 319 581 254 312 065 344, posting in NMBR-THRY Archives, 2008. Available at https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;f1ac1754.0803. |

[12] | B. C. Kellner, On a product of certain primes, J. Number Theory 179 (2017), 126-141. · Zbl 1418.11045 |

[13] | B. C. Kellner and J. Sondow, Power-sum denominators, Amer. Math. Monthly 124 (2017), 695-709. · Zbl 1391.11052 |

[14] | B. C. Kellner and J. Sondow, The denominators of power sums of arithmetic progressions, Integers 18 (2018), #A95, 1-17. · Zbl 1423.11029 |

[15] | B. C. Kellner and J. Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 1-21. · Zbl 1479.11028 |

[16] | A. Korselt, Problème chinois, L’Intermédiaire Math. 6 (1899), 142-143. |

[17] | R. G. E. Pinch, The Carmichael numbers up to 10 21 , Proceedings of Conference on Algo-rithmic Number Theory 2007, A. Ernvall-Hytönen et al., eds., TUCS General Publication 46, Turku Centre for Computer Science, 2007, 129-131. |

[18] | R. G. E. Pinch, The Carmichael numbers up to 10 18 , 2008. Available at http://www.s369624816.websitehome.co.uk/rgep/cartable.html. · Zbl 0780.11069 |

[19] | A. M. Robert, A Course in p-adic Analysis, GTM 198, Springer-Verlag, New York, 2000. · Zbl 0947.11035 |

[20] | J. H. Silverman, Taxicabs and sums of two cubes, Amer. Math. Monthly 100 (1993), 331-340. · Zbl 0796.11022 |

[21] | N. J. A. Sloane, ed., The On-Line Encyclopedia of Integer Sequences, http://oeis.org. · Zbl 1044.11108 |

[22] | S. S. Wagstaff, Jr., Large Carmichael numbers, Math. J. Okayama Univ. 22 (1980), 33-41. · Zbl 0427.10010 |

[23] | D. W. Wilson, The fifth taxicab number is 48 988 659 276 962 496, J. Integer Seq. 2 (1999), Article 99.1.9. · Zbl 0957.11042 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.