Divisibility and powers

Number Theory Level 3

Let p p be a prime number greater than 2.

What is the smallest value of p p such that 2 p − 1 2^p - 1 and 2 p + 1 2^p + 1 are both composite?


The answer is 11.

Denton Young by Denton Young , 47, USA

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1 solution

Denton Young
Nov 11, 2016

For p = 3, we have 7 and 9. 7 is prime.

For p = 5, we have 31 and 33. 31 is prime.

For p = 7, we have 127 and 129. 127 is prime.

For p = 11, we have 2047 and 2049. 2047 = 23 * 89, and 2049 = 3 * 683.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Is this solvable without trial and error?

Pi Han Goh - 4 years, 7 months ago

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Well, we can start with the fact that for an odd number, 2 p + 1 2^p + 1 is always divisible by 3. Then a look at the list of Mersenne primes would do the trick.

Denton Young - 4 years, 7 months ago

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