(i) It is sensible to start by trying out some small values for a and b.

If a + b is prime, then a cannot equal b, and if 2a – 1 and 2b – 1 are to be prime, neither a nor b can be 1. Also, exactly one of a and b must be even, for otherwise a + b is even, and therefore not prime.

Try a = 2, b = 3: then both a and b are prime (but 2a – 1 = 3, 2b – 1= 5 and a + b = 5, which could be useful to know).

Try a = 3, b = 4: then 2a – 1 = 5, 2b – 1= 7 and a + b = 7, so this can be our example for (i)

(ii) If we want both a and b to be prime then one of them must be 2, in order for a + b to be a prime bigger than 2.

This means we must consider pairs 2, b with all of b, 2b – 1 and b + 2 being prime.

We have already seen that b = 3 is a possibility.

If b and b + 2 are both primes greater than 3, then they must be either 1 more or 1 less than a multiple of 6.

(If they are 2 more or less than a multiple of 6 they must be even and not equal to 2, so not prime. If they are 3 more or less than a multiple of 6 they are either 3 or a multiple of 3, so cannot be prime if they are greater than 3. There are no other options!)

Therefore we must have (for some integer k> 0), b = 6k - 1, in order that b + 2 = 6k + 1.

Now this means that 2b – 1 = 2(6k - 1) – 1 = 12k – 3 = 3(4k – 1), which is divisible by 3.

So it is not possible for 2b – 1 to be prime if b is a prime greater than 3.

Hence, the only possible values of a and b with this property, and with a and b also prime, is

a = 2, b = 3.

Challenge 336: Producing Primes