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If a and n are relatively prime so that the greatest common divisor GCD(a,n)=1, then a^(lambda(n))=1 (mod n), where lambda is the Carmichael function.

The Cramér conjecture is the unproven conjecture that lim sup_(n->infty)(p_(n+1)-p_n)/((lnp_n)^2)=1, where p_n is the nth prime.

The distinct prime factors of a positive integer n>=2 are defined as the omega(n) numbers p_1, ..., p_(omega(n)) in the prime factorization ...

A number D that possesses no common divisor with a prime number p is either a quadratic residue or nonresidue of p, depending whether D^((p-1)/2) is congruent mod p to +/-1.

A finite field is a field with a finite field order (i.e., number of elements), also called a Galois field. The order of a finite field is always a prime or a power of a ...

A primitive subgroup of the symmetric group S_n is equal to either the alternating group A_n or S_n whenever it contains at least one permutation which is a q-cycle for some ...

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 ...