AP Notes, Outlines, Study Guides, Vocabulary, Practice Exams and more!

Number theory

Zsigmondy

Subject: 
Calculus
Rating: 
0
No votes yet
SocialTags: 
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
Subscribe to RSS - Number theory

Need Help?

We hope your visit has been a productive one. If you're having any problems, or would like to give some feedback, we'd love to hear from you.

For general help, questions, and suggestions, try our dedicated support forums.

If you need to contact the Course-Notes.Org web experience team, please use our contact form.

Need Notes?

While we strive to provide the most comprehensive notes for as many high school textbooks as possible, there are certainly going to be some that we miss. Drop us a note and let us know which textbooks you need. Be sure to include which edition of the textbook you are using! If we see enough demand, we'll do whatever we can to get those notes up on the site for you!