tailieunhanh - Lifting the exponent - Version 6

Lifting The Exponent Lemma is a powerful method for solving exponential Diophantine equations. It is pretty well-known in the Olympiad folklore though its origins are hard to trace. Mathematically, it is a close relative of the classical Hensel’s lemma in number theory (in both the statement and the idea of the proof). In this article we analyze this method and present some of its applications. | Lifting The Exponent Lemma LTE Version 6 - Amir Hossein Parvardi April 7 2011 Lifting The Exponent Lemma is a powerful method for solving exponential Diophantine equations. It is pretty well-known in the Olympiad folklore see . 3 though its origins are hard to trace. Mathematically it is a close relative of the classical Hensel s lemma see 2 in number theory in both the statement and the idea of the proof . In this article we analyze this method and present some of its applications. We can use the Lifting The Exponent Lemma this is a long name let s call it LTE in lots of problems involving exponential equations especially when we have some prime numbers and actually in some cases it explodes the problems . This lemma shows how to find the greatest power of a prime p -which is often 3 - that divides an bn for some positive integers a and b. The proofs of theorems and lemmas in this article have nothing difficult and all of them use elementary mathematics. Understanding the theorem s usage and its meaning is more important to you than remembering its detailed proof. I have to thank Fedja darij grinberg Darij Grinberg makar and ZetaX Daniel for their notifications about the article. And I specially appreciate JBL Joel and Fedja helps about TeX issues. 1 Definitions and Notation For two integers a and b we say a is divisible by b and write b a if and only if there exists some integer q such that a qb. We define vp x to be the greatest power in which a prime p divides x in particular if vp x a then pa x but pa 1 ị x. We also write pa x if and only if vp x a. So we have vp xy vp x vp y and vp x y min vp x Vp y . Example. The greatest power of 3 that divides 63 is 32. because 32 9 63 but 33 27 ị 63. in particular 321 63 or V3 63 2. Example. Clearly we see that if p and q are two different prime numbers then vp paq a or pa paq . Note. We have vp 0 TO for all primes p. 1 2 Two Important and Useful Lemmas Lemma 1. Let x and y be not necessary positive integers and let n

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