Can Prime Numbers Be Arbitrarily Far Apart?

💡 The Answer

• For any positive integer n, there exists a sequence of n consecutive composite numbers.
• This is proven by constructing the sequence (n+1)!+2 through (n+1)!+(n+1).
• Each term is divisible by numbers from 2 to n+1, so none is prime.
• Thus prime gaps can be as large as we want, though small gaps also occur infinitely often.