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^[1] 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.^[2] 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 $\mathcal{Z}(W_n)$.^[3]

See also

• Carmichael's theorem

References

1. Y. Bilu, G. Hanrot, P.M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew. Math. 539 (2001), 75-122
2. J.H. Silverman, Wieferich's criterion and the abc-conjecture, J. Number Theory 30 (1988), 226-237
3. P. Ingram, J.H. Silverman, Uniform estimates for primitive divisors in elliptic divisibility sequences, Number theory, Analysis and Geometry, Springer-Verlag, 2010, 233-263.

• K. Zsigmondy (1892). "Zur Theorie der Potenzreste". Journal Monatshefte für Mathematik 3 (1): 265–284. doi:10.1007/BF01692444.
• Th. Schmid (1927). "Karl Zsigmondy". Jahresbericht der Deutschen Mathematiker-Vereinigung 36: 167–168.
• Moshe Roitman (1997). "On Zsigmondy Primes". Proceedings of the American Mathematical Society 125 (7): 1913–1919. doi:10.1090/S0002-9939-97-03981-6. JSTOR 2162291.
• Walter Feit (1988). "On Large Zsigmondy Primes". Proceedings of the American Mathematical Society (American Mathematical Society) 102 (1): 29–36. doi:10.2307/2046025. JSTOR 2046025.
• Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence sequences. Mathematical Surveys and Monographs 104. Providence, RI: American Mathematical Society. pp. 103–104. ISBN 0-8218-3387-1. Zbl 1033.11006.

External links

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