Finding large prime gaps without using brute force

Question

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?