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n divides a^n-a for all integers a iff n is squarefree and (p-1)|(n-1) for all prime divisors p of n. Carmichael numbers satisfy this criterion.

If the first case of Fermat's last theorem is false for the prime exponent p, then 3^(p-1)=1 (mod p^2).

A positive integer n is kth powerfree if there is no number d such that d^k|n (d^k divides n), i.e., there are no kth powers or higher in the prime factorization of n. A ...

A primary ideal is an ideal I such that if ab in I, then either a in I or b^m in I for some m>0. Prime ideals are always primary. A primary decomposition expresses any ideal ...

When the group order h of a finite group is a prime number, there is only one possible group of group order h. Furthermore, the group is cyclic.

A knot is called prime if, for any decomposition as a connected sum, one of the factors is unknotted (Livingston 1993, pp. 5 and 78). A knot which is not prime is called a ...

An Abelian planar difference set of order n exists only for n a prime power. Gordon (1994) has verified it to be true for n<2000000.

Let a!=b, A, and B denote positive integers satisfying (a,b)=1 (A,B)=1, (i.e., both pairs are relatively prime), and suppose every prime p=B (mod A) with (p,2ab)=1 is ...

A ring for which the product of any pair of ideals is zero only if one of the two ideals is zero. All simple rings are prime.

Given an ideal A, a semiprime ring is one for which A^n=0 implies A=0 for any positive n. Every prime ring is semiprime.