Prove that $$(n+1)$$ is either itself a prime or a product of primes for every positive integer $$n$$.

We prove by strong induction that every integer $$N \ge 2$$ is a prime or a product of primes; taking $$N = n + 1$$ gives the result. For $$N = 2$$ the statement holds since 2 is prime. Suppose it holds for all integers from 2 to $$N - 1$$. If $$N$$ is prime we are done; otherwise $$N = uv$$ with $$2 \le u, v \le N - 1$$, and by hypothesis $$u$$ and $$v$$ are each a prime or a product of primes, so $$N = uv$$ is a product of primes. This completes the induction.