Question

Call the prime numbers, in order, p1, p2, p3 and so on. Thus p1 = 2, p2 = 3, p3 = 5, ...

(ii) Explain why p1p2p3 ... pn +1 is either prime or has a prime factor greater than pn for all positive integers of n.

(iii) How does that allow you to prove that there is an infinite number of prime numbers?

Answer 1

(i) Let the product of the first n prime numbers be N, then the remainder when N+1 is divided by any of the prime factors of N must be 1. p₁=2 and p₂=3, so p₂>2, its subscript. Similarly p₃=5 and p₃>3 its subscript. p₄=7>4 and p₄=p₁×p₂+1. If N+1 is prime, its index must be greater than the index of the largest prime factor of N. If N+1=pᵪ then N+1>χ. If N+1 is a composite, none of its factors can be included in the set of the first n prime numbers, because each of these gives a remainder of 1 when divided into N+1.

(ii) Let the product of the first χ prime numbers be N', where pᵪ is as defined in (i), then we know from (i) that N'+1 must be either a prime itself, or be a composite with a factor greater than pᵪ. That means there is another prime pᵧ>pᵪ. This implies that we can never find an ultimate prime number, because we can either generate a larger prime directly through prime products, or indirectly, because a generated composite has a prime factor greater than the largest we have generated so far. The implication is that the number of primes is infinite.