In elementary number theory, the lifting-the-exponent lemma provides several formulas for computing the p-adic valuation ν p {\displaystyle \nu _{p}} of binomial expressions which are differences of powers of integers: that is, how many factors of a prime p {\displaystyle p} are present. The lemma describes the steps necessary to "lift" the exponent of p {\displaystyle p} in such expressions. It is related to Hensel's lemma. It is often used in mathematical olympiads.
History By 1878, some ideas in the lemma had appeared in the work of mathematician Édouard Lucas, who was then a professor at Lycée Charlemagne. Lucas described related divisibility results (with a minor error in the case p = 2 {\displaystyle p=2} ). In 2006, Romanian mathematician Mihai Manea first published the modern and systematic formulation of the lemma, especially in the context of olympiad mathematics. By 2011, the lemma had become well-known in the math olympiad folklore, particularly through its use on mathematics forums such as the Art of Problem Solving.
Statements For a prime number p {\displaystyle p} , let ν p ( x ) = k {\displaystyle \nu _{p}(x)=k} , where p k {\displaystyle p^{k}} is the highest power of p {\displaystyle p} which is a divisor of x {\displaystyle x} , so that p k ∣ x , p k + 1 ∤ x {\displaystyle p^{k}\mid x,\ \ p^{k+1}\nmid x} . For any integers x {\displaystyle x} and y {\displaystyle y} with p ∤ x {\displaystyle p\nmid x} and p ∤ y {\displaystyle p\nmid y} , and a positive integer n {\displaystyle n} , the following statements hold:
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