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:

• n = 1, a − b = 1; then aⁿ − bⁿ = 1 which has no prime divisors
• n = 2, a + b a power of two; then any odd prime factors of a² - b² = (a + b)(a¹ - b¹) must be contained in a¹ - b¹, which is also even
• n = 6, a = 2, b = 1; then a⁶ − b⁶ = 63 = 3²7 = (a² − b²)²(a³ − b³)

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.

History

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

Generalizations

Let $(a_n)_{n\ge1}$ be a sequence of nonzero integers. The Zsigmondy set associated to the sequence is the set

$\mathcal{Z}(a_n) = \{ n \ge 1 : a_n \text{ has no primitive prime divisors} \}.$

i.e., the set of indices $n$ such that every prime dividing $a_n$ also divides some $a_m$ for some $m < n$. Thus Zsigmondy's theorem implies that $\mathcal{Z}(a^n-b^n)\subset\{1,2,6\}$, and Carmichael's theorem says that the Zsigmondy set of the Fibonacci sequence is $\{1,2,6,12\}$, and that of the Pell sequence is $\{1\}$. In 2001 Bilu, Hanrot, and Voutier proved that in general, if $(a_n)_{n\ge1}$ is a Lucas sequence or a Lehmer sequence, then $\mathcal{Z}(a_n) \subseteq \{ 1 \le n \le 30 \}$. Lucas and Lehmer sequences are examples of divisibility sequences.

It is also known that if $(W_n)_{n\ge1}$ is an elliptic divisibility sequence, then its Zsigmondy set $\mathcal{Z}(W_n)$ is finite. However, the result is ineffective in the sense that the proof does give an explicit upper bound for the largest element in $\mathcal{Z}(W_n)$, although it is possible to give an effective upper bound for the number of elements in $\mathcal{Z}(W_n)$.

See also

• Carmichael's theorem

External links

• Weisstein, Eric W., "Zsigmondy Theorem", MathWorld.