Zsigmondy's theorem

In number theory, Zsigmondy's theorem, named after Karl Zsigmondy, states that if ${\displaystyle a>b>0}$ are coprime integers, then for any integer ${\displaystyle n\geq 1}$, there is a prime number p (called a primitive prime divisor) that divides ${\displaystyle a^{n}-b^{n}}$ and does not divide ${\displaystyle a^{k}-b^{k}}$ for any positive integer ${\displaystyle k<n}$, with the following exceptions:

• ${\displaystyle n=1}$, ${\displaystyle a-b=1}$; then ${\displaystyle a^{n}-b^{n}=1}$ which has no prime divisors
• ${\displaystyle n=2}$, ${\displaystyle a+b}$ a power of two; then any odd prime factors of ${\displaystyle a^{2}-b^{2}=(a+b)(a^{1}-b^{1})}$ must be contained in ${\displaystyle a^{1}-b^{1}}$, which is also even
• ${\displaystyle n=6}$, ${\displaystyle a=2}$, ${\displaystyle b=1}$; then ${\displaystyle a^{6}-b^{6}=63=3^{2}\times 7=(a^{2}-b^{2})^{2}(a^{3}-b^{3})}$

This generalizes Bang's theorem,^[1] which states that if ${\displaystyle n>1}$ and ${\displaystyle n}$ is not equal to 6, then ${\displaystyle 2^{n}-1}$ has a prime divisor not dividing any ${\displaystyle 2^{k}-1}$ with ${\displaystyle k<n}$.

Similarly, ${\displaystyle a^{n}+b^{n}}$ has at least one primitive prime divisor with the exception ${\displaystyle 2^{3}+1^{3}=9}$.

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]