What integers are coprime to the first $x$ prime numbers? I have noticed that there is a very specific pattern to numbers that are coprime to $2$, it is simply all of the odd numbers.  More specifically, it is in the following pattern, where $n$ is an integer:
$$2n+1$$
For numbers that are coprime to the first $2$ prime numbers, $2$ and $3$ (or $6$), it is given as:
$$6n\pm1$$
And upon analyzing this, I found an equivalent expression:
$$6n\pm p_i$$
Where $p_i$ is the $i^{th}$ prime number greater than $2$ or $3$.  You can check this using modular arithmetic.
When considering a pattern to numbers that are coprime to the first $3$ prime numbers, I get the following:
$$30n\pm p_i$$
In fact, I have found that there is a particular $p_i$ which satisfies the entire problem for each problem, as we see $5$ satisfies $p_i$ for the first $2$ prime numbers problem.  In fact, $6n\pm5$ will never produce numbers that aren't coprime to the first $2$ prime numbers and never misses any numbers that are.
For the first 3 prime numbers, I note the following:
$$15n\pm2^a$$
For any integer $n$ or $a$, $15n\pm2^a$ will always be coprime to $2,3,5$ and there is no number it misses.
And then going back, I realize this also implies $3n\pm2^a$ is also coprime to $2$ and $3$.
In fact, I found that the following two expressions represent all numbers that are coprime to the first $x$ prime numbers:
$$\Pi_{v=1}^xp_vn\pm p_i$$
$$\frac{\Pi_{v=1}^xp_v}2n\pm2^a$$
And the second one reduces since $p_1=2$:
$$\Pi_{v=2}^xp_vn\pm2^a$$
Though I cannot construct a proof as to why such occurs.  Could someone explain why these two expressions manage to produce all numbers that are coprime to the first $x$ prime numbers and no numbers that aren't coprime?
 A: $\prod_{v=1}^x p_vn\pm p_i$ is certainly coprime to $p_1,\ldots, p_x$ as long as $p_i$ is not among these primes. On the other hand, Dirichlet showed that in each arithmetic progression $an+b$ with $a$, $b$ coprime, there are infinitely many primes. Therefore, we are sure to find enough distinct primes $p_i$ to civer all coprime residue classes. In fact, it suffices to consider only $\prod_{v=1}^x p_vn- p_i$ (or, if we allow negative $n$, only $\prod_{v=1}^x p_vn+ p_i$). Note that your observation "one prime suffices" does not generalize: There are $\prod_{v=1}^x (p_v-1)$ co prime residues classes and this is $>2$ as soon as we go beyond the first two primes.
Your second expression, $\frac12\prod_{v=1}^x p_vn\pm 2^a$ is a bit more problematic: If $n$ is a multiple of $4$, then the expression will be even!
But in all other cases, this is both odd and coprime to the remaining of the first $x$ primes because for each of these one summand is a multiple and the other is not. Hence with the caveat about $4\mid n$, this expression covers only coprime residue classes.
But does the expression cover all coprime residue classes? Perhaps surprisingly, it doesn't!
As $2^5\equiv 1\pmod{31}$, we see that the powers of $2$ run only over five distinct residue classes modulo $31$. If we add the negatives, $-2^a$, this count doubles to ten. But there are $30$ residue classes we want to cover.
A: That's just a coincidence, and it will vanish as soon as you'll go a bit further. Indeed, 30 is the greatest number with the property that all numbers below it and coprime to it are prime (or 1).
