# The Mysterious Case of the Hidden Evens

Find all positive integers n for which 3n−4, 4n−5, and 5n−3 are all prime numbers.

- 5 August, 2012 -
- Article, Math Problems -
- Tags : 104 problems, number theory, titu
- 0 Comments

We’re asked to find all integers satisfying are all primes.

The hidden trick behind this problem is that is even. Now, suppose that the sum of three prime integers turned out to be even, let’s enumerate all the ways to add three numbers to find out how this is possible.

So either exactly one of is even or that all of are even.

Let’s look at the case in which each of the primes is even. The only even prime number that I’m aware of is 2, so that means

This obviously has no solutions because is impossible, hence this case cannot hold.

Therefore, it must be the case that exactly one of is even.

Since was already shown to be impossible, then either

So both and/or .

Let’s first check if is satisfiable:

Hence is one solution.

Now, let’s check whether holds:

Therefore, only satisfies are all prime numbers.