Prime Gap Size is Unbounded
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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 $5$