Prime Gaps Are Unbounded: Factorial Proof for Any Size

Prime numbers are considered the fundamental building blocks of numbers, divisible only by one and themselves. All other non-prime numbers are formed by multiplying prime numbers together. There is an infinite number of primes, a fact that can be proven. While much discussion revolves around "prime gaps"—the spaces between consecutive primes, such as twin primes (primes with a gap of two)—a less explored question is how large these gaps can be.

The Unbounded Nature of Prime Gaps

The largest gap between two prime numbers is unbounded, meaning it can be arbitrarily large. While it cannot be infinite in the sense of a numerical value, the gap can be of any size desired. This concept can be demonstrated through a constructive proof, where a sequence of non-prime (composite) numbers of any specified length can be generated.

Constructing a Gap of Any Size

To illustrate this, consider the task of finding a gap of a specific size, for example, a gap of nine consecutive non-prime numbers.

1. Choose a number n: Let n be the desired number of consecutive non-primes. In this example, n = 9.
2. Calculate (n + 1)!: Compute the factorial of n + 1. For n = 9, this is (9 + 1)! = 10! = 3,628,800.

3. Generate the sequence: Create a sequence of numbers starting from (n + 1)! + 2 up to (n + 1)! + (n + 1).

   For n = 9, the sequence would be: * 10! + 2 * 10! + 3 * 10! + 4 * 10! + 5 * 10! + 6 * 10! + 7 * 10! + 8 * 10! + 9 * 10! + 10

Why This Method Works

The key to this construction lies in the properties of factorials:

• (n + 1)! is divisible by all integers from 1 to (n + 1): By definition, `(n + 1)! = 1
• 2
• 3
• ...
• (n + 1)`.
• (n + 1)! + k is composite if k is a factor of (n + 1)!:
  • Consider (n + 1)! + 2. Since (n + 1)! contains 2 as a factor (for n + 1 >= 2), both (n + 1)! and 2 are divisible by 2. Therefore, their sum, (n + 1)! + 2, is also divisible by 2, making it a composite number.
  • Similarly, (n + 1)! + 3 is divisible by 3 (for n + 1 >= 3), (n + 1)! + 4 is divisible by 4 (for n + 1 >= 4), and so on, up to (n + 1)! + (n + 1).

This method guarantees a list of n consecutive composite numbers. The first number in the sequence, (n + 1)! + 1, is not guaranteed to be composite, which is why the sequence starts from + 2. By starting with (n + 1)! + 2, we ensure that all n numbers in the generated sequence are composite.

Finding the Smallest Gaps

While the factorial method guarantees a gap of any size, it typically produces very large numbers. The generated sequence is almost certainly not the lowest example of a gap of that size. Finding the lowest example of a run of n consecutive composite numbers is a much more difficult problem.

For instance, a list of nine consecutive non-primes can be found starting at 114. This is significantly smaller than the numbers generated by 10! + k. There's also a famous run of seven non-primes starting at 90, which is the longest such run under 100.

Implications for Prime Number Search

The existence of arbitrarily large prime gaps has implications for finding large prime numbers. If one attempts to find a prime number by testing consecutive integers (e.g., starting from a million, then a million and one, etc.), there's a possibility of landing in the middle of a very large prime gap. This means one might have to check many numbers before encountering the next prime. While computers can perform these checks quickly, the theoretical understanding of how frequently these large gaps occur is still developing.