bibkey: howe2000higherorder authors: Everett W. Howe year: 2000 title: “Higher-order Carmichael numbers” doi: 10.1090/S0025-5718-00-01225-4 claim: “Theorem 1 characterizes Carmichael numbers of order m by squarefreeness and n congruent to a power of p modulo p^r-1 for every p dividing n and 1<=r<=m.” strata_touched: [] license: citation-only triage: anchor

Endomorphism, identity, and affine-shift tests

Primary source: Mathematics of Computation 69 (2000), 1711–1719. Author preprint: https://arxiv.org/abs/math/9812089v1.

Theorem 1 concerns exponentiation by n as an endomorphism on every Z/nZ-algebra generated by at most m elements as a module. For composite n, it is equivalent to n being squarefree and, for each p|n and 1≤r≤m, the existence of an integer i≥0 with n ≡ p^i (mod p^r-1).

The paper separately defines rigid Carmichael numbers, for which this endomorphism is the identity on every such finite étale algebra. The identity condition is stronger: on a field of order p^r it requires p^r-1 | n-1.

The infinitude discussion in this paper is heuristic, not a theorem of infinitude in every order. Grantham’s fixed-splitting-field theorem has a different quantifier domain and must be cited separately.

Testing only (X+a)^n-X^n-a probes an affine family of elements in one algebra. It does not assert the identity on that algebra, nor the endomorphism property on every algebra of bounded module rank. For example, exponentiation by a prime p is a nontrivial automorphism on F_(p^r) when r>1, while all these affine-shift identities still hold.