Zsigmondy's Theorem

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

In number theory, Zsigmondy's theorem, named after Karl Zsigmondy, states that if a > b > 0 {\displaystyle a>b>0} are coprime integers, then for any integer n ≥ 1 {\displaystyle n\geq 1} , there is a prime number p (called a primitive prime divisor) that divides a n − b n {\displaystyle a^{n}-b^{n}} and does not divide a k − b k {\displaystyle a^{k}-b^{k}} for any positive integer k < n {\displaystyle k<n} , with the following exceptions: n = 1 {\displaystyle n=1} , a − b = 1 {\displaystyle a-b=1} ; then a n − b n = 1 {\displaystyle a^{n}-b^{n}=1} which has no prime divisors n = 2 {\displaystyle n=2} , a + b {\displaystyle a+b} a power of two; then any odd prime factors of a 2 − b 2 = ( a + b ) ( a 1 − b 1 ) {\displaystyle a^{2}-b^{2}=(a+b)(a^{1}-b^{1})} must be contained in a 1 − b 1 {\displaystyle a^{1}-b^{1}} , which is also even n = 6 {\displaystyle n=6} , a = 2 {\displaystyle a=2} , b = 1 {\displaystyle b=1} ; then a 6 − b 6 = 63 = 3 2 × 7 = ( a 2 − b 2 ) 2 ( a 3 − b 3 ) {\displaystyle a^{6}-b^{6}=63=3^{2}\times 7=(a^{2}-b^{2})^{2}(a^{3}-b^{3})} This generalizes Bang's theorem, which states that if n > 1 {\displaystyle n>1} and n {\displaystyle n} is not equal to 6, then 2 n − 1 {\displaystyle 2^{n}-1} has a prime divisor not dividing any 2 k − 1 {\displaystyle 2^{k}-1} with k < n {\displaystyle k<n} .

In number theory, Zsigmondy's theorem, named after Karl Zsigmondy, states that if a > b > 0 are coprime integers, then for any int