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 (called a primitive prime divisor) 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. History
The theorem was discovered by Zsigmondy working in Vienna from 1894 until 1925. Generalizations 
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  (see , there are only 13 such s, namely 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 18, 30). 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 . See also
 Carmichael's theorem

References

External links 

Theorems in number theory