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Integer sequences

Zsigmondy

Subject: 
Calculus
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Fields of mathematics
Mathematics
Algebraic number theory
Number theory
Integer sequences
Zsigmondy's theorem
Factorization
Prime number
XTR
Fermat's Last Theorem
Coprime integers
Fermat's theorem on sums of two squares

The Zsigmondy Theorem PISOLVE 7/31/11 Abstract In this paper we will be discussing the uses and applications of Zsigmondy?s Theorem. As some of you may consider the theorem as ?large/ brutal? , it can actually be proved by elementary methods [1] - at the same time, it is applicable in so many Number Theoretic problems(All problems in this article are sourced in AoPS). We see no reasons not to use it. Intro Problem Let p > 3 be a prime. Show that every positive divisor of 2 p+1 3 is in the form 2kp+ 1. Solution: We show that all prime divisors are in this form, then the result readily follows.Let q| 2 p+1 3 . Then: q|22p ? 1 o2(q)|2p If o2(q) 6= 2p, we have 3 cases: o2(q) = 1 Then q|1, clearly impossible. o2(q) = 2
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