On prime divisors of differences two nth powers
In number theory, Zsigmondy's theorem, named after Karl Zsigmondy, states that if formula_1 are coprime integers, then for any integer formula_2, there is a prime number "p" (called a "primitive prime divisor") that divides formula_3 and does not divide formula_4 for any positive integer formula_5, with the following exceptions:
This generalizes Bang's theorem, which states that if formula_17 and formula_18 is not equal to 6, then formula_19 has a prime divisor not dividing any formula_20 with formula_5.
Similarly, formula_22 has at least one primitive prime divisor with the exception formula_23.
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 formula_24 be a sequence of nonzero integers.
The Zsigmondy set associated to the sequence is the set
formula_25
i.e., the set of indices formula_18 such that every prime dividing formula_27 also divides some formula_28 for some formula_29. Thus Zsigmondy's theorem implies that formula_30, and Carmichael's theorem says that the Zsigmondy set of the Fibonacci sequence is formula_31, and that of the Pell sequence is formula_32. In 2001 Bilu, Hanrot, and Voutier
proved that in general, if formula_24 is a Lucas sequence or a Lehmer sequence, then formula_34 (see OEIS: , there are only 13 such formula_18s, namely 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 18, 30).
Lucas and Lehmer sequences are examples of divisibility sequences.
It is also known that if formula_36 is an elliptic divisibility sequence, then its Zsigmondy
set formula_37 is finite. However, the result is ineffective in the sense that the proof does not give an explicit upper bound for the largest element in formula_37,
although it is possible to give an effective upper bound for the number of elements in formula_37.
References.
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