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Number Theory Level 3

Find the largest integer n n such that 2 n 2^n divides 3 1024 − 1 3^{1024} - 1 .


The answer is 12.

Prokash Shakkhar by Prokash Shakkhar , 22, Bangladesh

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2 solutions

Reynan Henry
Dec 29, 2016

It is a direct use of Lifting the exponent theorem with p = 2 p=2 \\

v 2 ( 3 2014 − 1 ) = v 2 ( 3 − 1 ) + v 2 ( 3 + 1 ) + v 2 ( 1024 ) − 1 = 1 + 2 + 10 − 1 = 12 v_2(3^{2014}-1)=v_2(3-1)+v_2(3+1)+v_2(1024)-1 = 1+2+10-1=12

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WolframAlpha input: . .

solve over integers mod(3^1024-1,2^n)=0

press "more solutions" until all values are listed, the last value is n=12 (the largest).

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