Lifting The Exponent Lemma Version 6 by Amir Hossein Parvardi

- Lifting The Exponent Lemma is a powerful method for solving exponential Diophantine equations. It is pretty well-known in the Olympiad folklore (see, e.g., [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.

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