Numbers for which Sixth Power plus 1091 is Composite

Theorem
The number $1091$ has the property that:
 * $x^6 + 1091$

is composite for all integer values of $x$ from $1$ to $3095$.

Proof
We have:

Hence $3096^6 + 1091$ is composite (divisible by $7$).

In fact it has prime factorization $7^2 \times 3203 \times 3020201 \times 1857876521$.

I have checked around $30$ numbers manually (using the results below) and verified the result:
 * $3906$ is the smallest $x$ such that $x^6 + 1091$ is prime.

It is simple to prove the partial result however:

Suppose $x^6 + 1091$ is prime.

Then:
 * $x$ is a multiple of $42$ ending in $0$, $4$ or $6$ in decimal notation
 * $x \equiv 0, \pm 2, \pm 5, \pm 6 \pmod {13}$
 * $x \not \equiv 4, 6, 9 \pmod {19}$

The proof is split into $6$ parts:

$x$ is a multiple of $2$
Suppose not. Then $x$ is odd, and so is $x^6$.

Hence $x^6 + 1091$ is even, and thus composite.

Thus we must require $x$ to be even.

$x$ is a multiple of $3$
Suppose not. Write $x = 3 k \pm 1$.

Hence:

showing that $x^6 + 1091$ is composite (divisible by $3$).

Thus we must require $x$ to be a multiple of $3$.

$x$ is a multiple of $7$
Suppose not. Then $x \perp 7$.

Then:

showing that $x^6 + 1091$ is composite (divisible by $7$).

Thus we must require $x$ to be a multiple of $7$.

$x$ must end in $0$, $4$ or $6$
From the above we require $x$ to be even.

Suppose $x$ ends in $2$ or $8$.

Then $x^6$ ends in $4$.

Thus $x^6 + 1091$ ends in $5$, which by Divisibility by 5 is divisible by $5$.

Hence we must require $x$ to end in $0$, $4$ or $6$.

$x \equiv 0, \pm 2, \pm 5, \pm 6 \pmod {13}$
Here is a table of $x^6 \pmod {13}$:


 * $\begin{array}{|c|c|c|c|c|c|c|c|}

\hline x \bmod {13} & 0 & \pm 1 & \pm 2 & \pm 3 & \pm 4 & \pm 5 & \pm 6 \\ \hline x^6 \bmod {13} & 0 & 1 & -1 & 1 & 1 & -1 & -1 \\ \hline \end{array}$

For $x \equiv \pm 1, \pm 3, \pm 4 \pmod {13}$:

showing that $x^6 + 1091$ is composite (divisible by $13$).

Thus we must require $x \equiv 0, \pm 2, \pm 5, \pm 6 \pmod {13}$.

A proof similar to above can show that $x \not \equiv 4, 6, 9 \pmod {19}$.