Question

The number of primes $p$ such that $p,p+10,p+14$ are all prime numbers is

• A. $0$
• B. $1$
• C. $3$
• D. infinitely many

Answer 1

The correct option is B $1$

If $p=3,$ then the numbers are $3,13,17.$ All of them are prime.
So, $p=3$ is a solution.

Now, any other prime $p$ is either of the form $3k+1$ or $3k+2.$

If $p=3k+1,$
then $p+14=3k+15=3(k+5)$ which is not prime.

If $p=3k+2,$
then $p+10=3k+12=3(k+2)$ which is not prime.

Thus, there is only such prime $p.$