Is 11 always a prime?

Number Theory Level 2

How many prime numbers $p$ exist, such that $11_p$ is also a prime?

If you think that there are infinitely many, enter $-1$ as your answer.

Details and assumptions

$n_m$ denotes that the number $n$ is written in base $m$ .

1 solution

Braden Dean
Dec 29, 2020

$11_p$ written in base 10 is equal to $p + 1$ .

Since every prime number except 2 is odd, $11_p$ must be even, and therefore composite, for any $p > 2$ .

For $p = 2$ :

$11_2 = 2_{10} + 1_{10} = 3_{10}$ .

As we know 3 is prime, we know $11_2$ is prime.

Therefore, there exists $\fbox{1}$ prime number for which this is true.