Maths
Is there a largest prime?
Is there a largest prime number? Convince me either way.
- Hint 1Suppose there were only finitely many primes. Multiply them all together.
- Hint 2What is the remainder when you divide that product plus one by any of the primes?
- Suppose the primes were only p₁, p₂, …, pₖ.
- Let N = p₁p₂…pₖ + 1. Dividing N by any pᵢ leaves remainder 1.
- N > 1, so it has a prime factor, and that factor cannot be any of the pᵢ.
- That contradicts the list being complete, so there is no largest prime.
Where it lands: No: there are infinitely many primes.
The trap: Claiming N itself must be prime. 2 × 3 × 5 × 7 × 11 × 13 + 1 = 30031 = 59 × 509. N only has a prime factor missing from the list.
What a tutor might ask next: Can you show there are infinitely many primes that leave remainder 3 when divided by 4?
Now do one out loud. In the interview you think aloud with a tutor. Try a Maths problem with hints, follow-ups and Mia's feedback on how you think.