UKMT Senior Challenge-XIII

Number Theory Level 2

A Mersenne prime is a prime of the form 2 p − 1 2^p-1 , where p p is also a prime. One of the following is not a Mersenne prime. Which one is it?

This problem is not original.This problem is part of this set .

2 2 − 1 2^2-1 2 3 − 1 2^3-1 2 5 − 1 2^5-1 2 11 − 1 2^{11}-1 2 7 − 1 2^7-1

A Former Brilliant Member by A Former Brilliant Member

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1 solution

Let us go through all the objective answer options.

p 2 p − 1 2 3 3 7 5 31 7 127 11 2047 \begin{array} {rr} p & 2^p-1 \\ 2 & 3 \\ 3 & 7 \\ 5 & 31 \\ 7 & 127 \\ 11 & 2047 \end{array}

2047 = 23 × 89 \boxed{2047} = 23 \times 89 is not a prime.

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