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 natural number n > 1 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:

This generalized 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, a^n %2B b^n has at least one primitive prime divisor with the exception 2^3 %2B 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

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History

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

See also

References

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