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 > 1 there is a prime number p (called a primitive prime divisor) that divides $$a^n − b^n$$ 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.

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

Similarly, $$a^n + b^n$$ has at least one primitive prime divisor with the exception $$2^3 + 1^3 = 9$$

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

Carmichael's theorem

References

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.[dead link]
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.
Graham Everest; Alf van der Poorten, Igor Shparlinski, Thomas Ward (2003). Recurrence sequences. Mathematical Surveys and Monographs. 104. American Mathematical Society. pp. 103–104. ISBN 0-8218-3387-1.