**Background:**
In running algebraic geometry computations using software such as Macaulay2, it is often easier and faster to work over $\mathbb F_p = \mathbb Z / p\mathbb Z$ for a large prime $p$, rather than over $\mathbb Q$. (Note that working directly over $\mathbb C$ is not really possible for exact computations.) Some basic model theory implies (more or less) that if you have a question that is capable of being answered by an algorithm, and the question has the same answer for $k$ as for $\bar{k}$, then its answer over $\mathbb Q$ will be the same as its answer over $\mathbb F_p$ for all but finitely many primes $p$. The accepted wisdom is that with virtually no exceptions, if you want to answer an algebro-geometric question over $\mathbb Q$, you can get a reliable answer by picking a large prime such as $p=32003$ and doing your computations over $\mathbb F_p$.

I think people generally pick a prime near the top of the range their software can handle, which is relatively easy to remember; $32003$ and $31667$, for instance, both fit the bill when using Macaulay2. However, I was wondering whether there are other mathematical characteristics of a prime that can affect how well it approximates characteristic zero. For instance, does some special pathology arise if $p$ is (or is not) a Mersenne prime, or if $(p-1)/2$ is (or is not) prime, or...?

Note that I only am asking about behavior that affects *how well characteristic $p$ approximates characteristic $0$.* Questions that one would only ever ask in finite characteristic are not relevant. Also irrelevant are questions that cannot be answered by an algorithm; for instance, "is $n\cdot1=0$ for some positive integer $n$" cannot be asked unless you include a bound on $n$.

**The Question:** Are there valid mathematical reasons for preferring one large prime over another to approximate characteristic $0$, other than the assumption that larger primes are generally better?