Carmichael numbers in number rings.

*(English)*Zbl 1176.11049A Carmichael number is a composite integer \(n\) such that \(a^n\equiv a\pmod n\) for all integers \(a\). It is known that there are infinitely many of them. Korselt’s criterion states that a composite integer \(n>1\) is Carmichael iff \(n\) is squarefree and \(p-1|n-1\) for all primes \(p|n\).

The purpose of this paper is to characterise Carmichael ideals. Let \(K\) be an extension of \(\mathbb Q\), and let \(I\) be a composite ideal in \(O_K\), the ring of integers of \(K\). The ideal \(I\) is Carmichael in \(O_K\) if for all \(\alpha \in O_K\) we have \[ \alpha^{N(I)}\equiv \alpha \bmod I, \] where \(N(I)=\#O_K/I\). The author establishes a generalised Korselt’s criterion. He also investigates when Carmichael numbers in the integers generate Carmichael ideals in the algebraic integers of an abelian field. Here he pays special attention to quadratic and cyclotomic fields. He also shows that if \(n\) is composite, then there exist infinitely many abelian number fields \(K\) such that \(\gcd(n, \text{Disc}(K))=1\) and \(n\) is not Carmichael in \(K\). There are further results that we cannot mention for reasons of space.\hfil

\indent The deepest ingredient in the proofs is a result due to Y. Bugeaud, P. Corvaja and U. Zannier [Math. Z. 243, No.1, 79–84 (2003; Zbl 1021.11001)] on an upper bound for the greatest common divisor of \(a^n-1\) and \(b^n-1\) and D. R. Heath-Brown’s [Q. J. Math., Oxf. II. Ser. 37, 27–38 (1986; Zbl 0586.10025)] classical result that all primes, with the possible exception of at most two, are primitive roots for infinitely many primes \(p\).

The purpose of this paper is to characterise Carmichael ideals. Let \(K\) be an extension of \(\mathbb Q\), and let \(I\) be a composite ideal in \(O_K\), the ring of integers of \(K\). The ideal \(I\) is Carmichael in \(O_K\) if for all \(\alpha \in O_K\) we have \[ \alpha^{N(I)}\equiv \alpha \bmod I, \] where \(N(I)=\#O_K/I\). The author establishes a generalised Korselt’s criterion. He also investigates when Carmichael numbers in the integers generate Carmichael ideals in the algebraic integers of an abelian field. Here he pays special attention to quadratic and cyclotomic fields. He also shows that if \(n\) is composite, then there exist infinitely many abelian number fields \(K\) such that \(\gcd(n, \text{Disc}(K))=1\) and \(n\) is not Carmichael in \(K\). There are further results that we cannot mention for reasons of space.\hfil

\indent The deepest ingredient in the proofs is a result due to Y. Bugeaud, P. Corvaja and U. Zannier [Math. Z. 243, No.1, 79–84 (2003; Zbl 1021.11001)] on an upper bound for the greatest common divisor of \(a^n-1\) and \(b^n-1\) and D. R. Heath-Brown’s [Q. J. Math., Oxf. II. Ser. 37, 27–38 (1986; Zbl 0586.10025)] classical result that all primes, with the possible exception of at most two, are primitive roots for infinitely many primes \(p\).

Reviewer: Pieter Moree (Bonn)

##### MSC:

11R04 | Algebraic numbers; rings of algebraic integers |

11A07 | Congruences; primitive roots; residue systems |

11A41 | Primes |

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\textit{G. A. Steele}, J. Number Theory 128, No. 4, 910--917 (2008; Zbl 1176.11049)

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##### References:

[1] | Alford, W.R.; Granville, Andrew; Pomerance, Carl, There are infinitely many Carmichael numbers, Ann. of math. (2), 139, 3, 703-722, (1994) · Zbl 0816.11005 |

[2] | Bugeaud, Y.; Corvaja, P.; Zannier, U., An upper bound for the G.C.D. of \(a^n - 1\) and \(b^n - 1\), Math. Z., 243, 1, 79-84, (2003) · Zbl 1021.11001 |

[3] | Heath-Brown, D.R., Artin’s conjecture for primitive roots, Quart. J. math. Oxford ser. (2), 37, 145, 27-38, (1986) · Zbl 0586.10025 |

[4] | Howe, Everett W., Higher-order Carmichael numbers, Math. comp., 69, 232, 1711-1719, (2000) · Zbl 0966.11006 |

[5] | Mirsky, L., The number of representations of an integer as the sum of a prime and a k-free integer, Amer. math. monthly, 56, 17-19, (1949) · Zbl 0033.16203 |

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