Zsigmondy's theorem

In number theory, Zsigmondy's theorem, named after Karl Zsigmondy, states that if are coprime integers, then for any integer, there is a prime number p that divides and does not divide for any positive integer, with the following exceptions:

, ; then which has no prime divisors
, a power of two; then any odd prime factors of must be contained in, which is also even
,, ; then

This generalizes Bang's theorem, which states that if and is not equal to 6, then has a prime divisor not dividing any with.
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. It is also used in Mathematical olympiads.

History

The theorem was discovered by Karl Zsigmondy who proved it in 1892.

Let be a sequence of nonzero integers.
The Zsigmondy set associated to the sequence is the set
i.e., the set of indices such that every prime dividing also divides some for some. Thus Zsigmondy's theorem implies that, and Carmichael's theorem says that the Zsigmondy set of the Fibonacci sequence is, and that of the Pell sequence is. In 2001 Bilu, Hanrot, and Voutier
proved that in general, if is a Lucas sequence or a Lehmer sequence, then .
Lucas and Lehmer sequences are examples of divisibility sequences.
It is also known that if is an elliptic divisibility sequence, then its Zsigmondy
set is finite. However, the result is ineffective in the sense that the proof does not give an explicit upper bound for the largest element in,
although it is possible to give an effective upper bound for the number of elements in.