It seems to me that there are three ways to find large prime gaps without using brute force.
- Natural Prime Order Using Factorials
For example, using $7!$, it is clear that $(7!+2), (7!+3), \dots (7!+8), (7!+9), (7!+10)$ are not primes.
Let $p_n$ be the $n$th prime. Around $p_n!+1$, there is a prime gap of at least length $p_{n+1}-1$.
This can be generalized to $a(b!)$ where $a \ge 1$ and $b \ge p_n$.
- Natural Prime Order Using Primorials
The same pattern for factorials holds for primorials.
For example, using $7\#$, it is clear that $(7\#+2), (7\#+3), \dots (7\#+8), (7\#+9), (7\#+10)$ are not primes.
This can be generalized to $a(b\#)$ where $a\ge1$ and $b\ge p_n$.
- Arbritrary Prime Order Using Chinese Remainder Theorem
For example, we can find a gap of at least $10$ using $4$ distinct primes say $2,3,7,11$.
This is equivalent to a CRT problem of finding $x$ where:
$x \equiv {-1} \pmod 2$
$x \equiv {-2} \pmod 3$
$x \equiv {-4} \pmod {11}$
$x \equiv {-6} \pmod 7$
Other numbers with this same prime gap can be found by adding factorials or primorials to $x$.
Here's my question:
Are these three methods the only way to find large prime gaps without using brute force? Are there any other methods that can be used?