In number theory, Zsigmondy's theorem, named after Karl Zsigmondy, states that if a > b > 0 are coprime integers, then for any natural number n, there is a prime number p (called a primitive prime divisor) that divides an − bn and does not divide ak − bk for any positive integer k < n, with the following exceptions:
- a - b = 1, and n = 1;
- a = 2, b = 1, and n = 6; or
- a + b is a power of two, and n = 2.
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 2k − 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 be a sequence of nonzero integers. The Zsigmondy set associated to the sequence is the set
i.e., the set of indices such that every prime dividing also divides some for some . Thus Zsigmondy's theorem implies that , and Carmichael's theorem says that the Zsigmondy set of the Fibonacci sequence is , and that of the Pell sequence is . In 2001 Bilu, Hanrot, and Voutier proved that in general, if is a Lucas sequence or a Lehmer sequence, then . Lucas and Lehmer sequences are examples of divisibility sequences.
It is also known that if is an elliptic divisibility sequence, then its Zsigmondy set is finite. However, the result is ineffective in the sense that the proof does give an explicit upper bound for the largest element in , although it is possible to give an effective upper bound for the number of elements in .
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- J.H. Silverman, Wieferich's criterion and the abc-conjecture, J. Number Theory 30 (1988), 226-237
- P. Ingram, J.H. Silverman, Uniform estimates for primitive divisors in elliptic divisibility sequences, Number theory, Analysis and Geometry, Springer-Verlag, 2010, 233-263.
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- 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.