Q. Three prime numbers p, q and r, each less than 20, are such that p – q = q – r. How many distinct possible values can we get for (p + q + r)?

Explanation : 

We seek all 3-term arithmetic progressions of primes below 20. If p, q, r are in arithmetic progression then

  p − q = q − r ⟹ p + r = 2q.

The primes under 20 are {2, 3, 5, 7, 11, 13, 17, 19}. Testing q in this set and looking for an integer step d>0 with q±d also prime, one finds exactly five progressions (up to order):

1. (3, 5, 7) sum = 15

2. (3, 7, 11) sum = 21

3. (5, 11, 17) sum = 33

4. (3, 11, 19) sum = 33

5. (7, 13, 19) sum = 39

The sums that occur are 15, 21, 33, 39. Hence there are 4 distinct values of p+q+r.