Zsigmondy's theorem

In number theory, Zsigmondy's theorem states that if "a" > "b" > 0 are coprime integers, then for any natural number "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:

*"a" = 2, "b" = 1, and "n" = 6; or

*"a" + "b" is a power of two, and "n" = 2.

History

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

References

*cite journal
title = Zur Theorie der Potenzreste
author = K. Zsigmondy
journal = Journal Monatshefte für Mathematik
volume = 3
issue = 1
pages = 265–284
year = 1892
doi = 10.1007/BF01692444
*cite journal
title = Karl Zsigmondy
author = Th. Schmid
journal = Jahresbericht der Deutschen Mathematiker-Vereinigung
volume = 36
issue =
pages = 167–168
year = 1927
url = http://www.digizeitschriften.de/no_cache/home/jkdigitools/loader/?tx_jkDigiTools_pi1%5BIDDOC%5D=517497
*cite journal
title = On Zsigmondy Primes
author = Moshe Roitman
journal = Proceedings of the American Mathematical Society
volume = 125
issue = 7
pages = 1913–1919
year = 1997
url = http://links.jstor.org/sici?sici=0002-9939%28199707%29125%3A7%3C1913%3AOZP%3E2.0.CO%3B2-2
doi = 10.1090/S0002-9939-97-03981-6
*cite journal
title = On Large Zsigmondy Primes
author = Walter Feit
journal = Proceedings of the American Mathematical Society
volume = 102
issue = 1
pages = 29–36
year = 1988
url = http://links.jstor.org/sici?sici=0002-9939%28198801%29102%3A1%3C29%3AOLZP%3E2.0.CO%3B2-B
doi = 10.2307/2046025
*cite book | author=Graham Everest | coauthors=Alf van der Poorten, Igor Shparlinksi, Thomas Ward | title=Recurrence sequences | series=Mathematical Surveys and Monographs | volume=104 | publisher=American Mathematical Society | year=2003 | isbn=0-8218-3387-1 | pages=103-104