Difference between revisions of "Lifting the Exponent"
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Let <math>p</math> be an odd prime, and let <math>a</math> and <math>b</math> be integers relatively prime to <math>p</math> such that <math>p \mid (a-b)</math>. Let <math>n</math> be a positive integer. Then the number of factors of <math>p</math> that divide <math>a^n - b^n</math> is equal to the number of factors of <math>p</math> that divide <math>a-b</math> plus the number of factors of <math>p</math> that divide <math>n</math>. | Let <math>p</math> be an odd prime, and let <math>a</math> and <math>b</math> be integers relatively prime to <math>p</math> such that <math>p \mid (a-b)</math>. Let <math>n</math> be a positive integer. Then the number of factors of <math>p</math> that divide <math>a^n - b^n</math> is equal to the number of factors of <math>p</math> that divide <math>a-b</math> plus the number of factors of <math>p</math> that divide <math>n</math>. | ||
Latest revision as of 18:59, 18 January 2024
(Lemma from MAA official solution, 2020 AIME I Problems/Problem 12)
Let 
be an odd prime, and let 
and 
be integers relatively prime to such that 
. Let 
be a positive integer. Then the number of factors of that divide 
is equal to the number of factors of that divide 
plus the number of factors of that divide .
