IMC 2009 Q16
16. How many different positive integers \(n\) are there for which \(n\) and \(n^3 + 3\) are both prime numbers?
- A 0
- B 1
- C 2
- D 3
- E infinitely many
Intermediate Mathematical Challenge 2009, Question 16. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 1
If \(n\) is an odd prime, then \(n^3 + 3\) is an even number greater than 3 and therefore not prime. The only even prime is 2 (which some would say makes it very odd!) and when \(n = 2\), \(n^3 + 3 = 11\) which is also prime. So there is exactly one value of \(n\) for which \(n\) and \(n^3 + 3\) are both prime.
Solution © UK Mathematics Trust, from the IMC 2009 solutions.