nt.number theory - Composite pairs of the form n!-1 and n!+1

As far as nonstandard models go: we can indeed get $mathbb{Z}$-like intervals $I$ such that each $xin I$ has a standard factor. The proof is via Compactness, and the Chinese Remainder Theorem:

First, adjoin a constant symbol $c$ to our language. Let $p_i$ be the $i^{th}$ prime number, let $q_i=p_{2i}$, and let $r_i=p_{2i+1}$.

Define numbers $a_i$, $b_i$ by recursion as follows:

$a_0=0$, $a_{n+1}=minlbrace x: forall kinmathbb{N}, jle n(cnot=a_j+kq_j)rbrace$

$b_0=0$, $b_{n+1}=minlbrace x: forall kinmathbb{N}, jle n(cnot=b_j+kr_j)rbrace$

Now, for each $iinmathbb{N}$, let $sigma_i$ express "$c$ is congruent to $-a_i$(mod$p_i$)", let $tau_i$ express "$c$ is congruent to $b_i$(mod$p_i$)," and let $Sigma=lbrace sigma_i: iinmathbb{N}rbracecuplbrace tau_i: iinmathbb{N}rbrace$. By the Chinese Remainder Theorem, every finite subset of $Sigma$ is consistent with True Arithmetic $TA$, so by Compactness, $Sigma$ itself is consistent with $TA$. So there is some nonstandard model of $TA$ in which $Sigma$ holds; clearly, in such a model, every number in the $mathbb{Z}$-like interval centered on $c$ has a standard factor.

I have no idea whether $every$ nonstandard model has such an interval, however.