Library / Berkeley Math Circle Monthly Contest 7 /7 of 15

Berkeley Math Circle Monthly Contest 7

Number theory Difficulty 4.8 AIME Prove it United States

Problem:

Find all positive prime numbers $p$ such that $p+2$ and $p+4$ are prime as well.

Solution

Solution:

For $p=3$ , $p+2=5$ , $p+4=7$ and these are obviously prime.

For $p>3$ , we know that $p$ is not divisible by $3$ . The remainder of $p$ when divided by $3$ can be either $1$ or $2$ . If it is $1$ , then $p+2$ is divisible by $3$ ; if it is $2$ , then $p+4$ is divisible by $3$ . Hence $p=3$ is the only solution.

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.