Prime Gap Size is Unbounded

Theorem

Let $n \in \N$ be a natural number, arbitrarily large.

Then there exist consecutive prime numbers $p_1$ and $p_2$ such that:

$p_2 - p_1 > n$

That is, the size of prime gaps is unbounded.

That is, there are blocks of consecutive composite numbers whose length exceeds any given $n \in \N$.


Proof

Let $p$ be a prime number greater than $n + 1$.

Consider the primorial:

$q = 2 \times 3 \times 5 \times \cdots \times p$

All of the $p - 1$ numbers:

$q + 2, q + 3, q + 4, \ldots, q + p$

are composite.

Hence the result.

$\blacksquare$


Sources

  • 1979: G.H. Hardy and E.M. Wright: An Introduction to the Theory of Numbers (5th ed.) ... (previous) ... (next): $\text I$: The Series of Primes: $1.4$ The sequence of primes: Theorem $5$
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