A Huge Exponent

Number Theory Level 4

If m m is an integer such that 3 m ∣ 7 3 530 − 1 3^m|7^{3^{530}}-1 , what is the maximum possible value of m m ?


The answer is 531.

Alan Yan by Alan Yan , 20, USA

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1 solution

Alan Yan
Nov 9, 2015

Since 3 ∣ 7 − 1 3 | 7 - 1 , by Lifting the Exponent Lemma, we have v 3 ( 7 3 530 − 1 ) = v 3 ( 3 530 ) + v 3 ( 7 − 1 ) = 531 . v_3(7^{3^{530}} - 1) = v_3(3^{530}) + v_3(7-1) = \boxed{531}.

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