Zsigmondy's Theorem

In number theory, Zsigmondy's theorem, named after Karl Zsigmondy, states that if a > b > 0 are coprime integers, then for any natural number n > 1 there is a prime number p (called a primitive prime divisor) that divides an − bn and does not divide ak − bk for any positive integer k < n, with the following exceptions:

• a = 2, b = 1, and n = 6; or

• a + b is a power of two, and n = 2.

This generalized Bang's theorem which states that if n>1 and n is not equal to 6, then 2n-1 has a prime divisor not dividing any 2k-1 with k<n.

Similarly, has at least one primitive prime divisor with the exception

Zsigmondy's theorem is often useful, especially in group theory, where it is used to prove that various groups have distinct orders except when they are known to be the same