Zsigmondy's theorem

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

This generalizes Bang's theorem, which states that if n > 1 and n is not equal to 6, then 2ⁿ − 1 has a prime divisor not dividing any 2^k − 1 with k < n.

Similarly, aⁿ + bⁿ has at least one primitive prime divisor with the exception 2³ + 1³ = 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.

The theorem was discovered by Zsigmondy working in Vienna from 1894 until 1925.

Let ${\displaystyle (a_{n})_{n\geq 1}}$ be a sequence of nonzero integers. The Zsigmondy set associated to the sequence is the set

${\displaystyle {\mathcal {Z}}(a_{n})=\{n\geq 1:a_{n}{\text{ has no primitive prime divisors}}\}.}$

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