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find positive n so that exists prime p with p^n-(p-1)^n$ a power of 3

Source: JBMO Shortlist 2017 NT5

July 25, 2018

number theoryPowerpositive integerprime

Problem Statement

Find all positive integers

n

such that there exists a prime number

p

, such that

p^n-(p-1)^n

is a power of

3

.
Note. A power of

3

is a number of the form

3^a

where

a

is a positive integer.