The Zsigmondy Method

Zsigmondy's method is a mathematical technique used in number theory to analyze when a number of the form \(a^n - b^n\) has a new prime divisor that hasn't appeared in earlier similar differences. It helps identify "primitive prime divisors"—primes that only divide \(a^n - b^n\) at the specific exponent \(n\). This method is useful in understanding the factors of such expressions and has applications in areas like algebra, cryptography, and the proof of important mathematical theorems. Essentially, it provides a systematic way to predict when new prime factors will emerge in these types of number sequences.