The Zsigmondy theorem states that if 
and 
are positive integers
with 
,
is an integer,
and 
(i.e., 
and 
are relatively prime), then 
has at least one primitive
prime factor with the following two possible exceptions:
1. 
.
2. 
and 
is a power of 2.
Similarly, if 
are relatively prime integers
and 
is an integer, then 
has at least one primitive
prime factor with the exception 
.
A specific case of the theorem considers the 
th Mersenne number 
, then each of 
, 
, 
, ... has a prime factor
that does not occur as a factor of an earlier member of the sequence, except
for 
.
For example, the Zsigmondy numbers for 
, 
, 
, ... are 1, 3, 7, 5, 31, 1, 127, 17, 73, 11, 
, ... (OEIS A064078).
A value of 1 denotes the absence of a primitive
prime factor. These factors are sometimes called the Zsigmondy numbers 
.
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 (Montgomery 2001).
