On prime divisors of differences two nth powers
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,[1] 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.[2][3]