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Maybe this question looks a little vague, but I was doing something else when it hit me. I will delete this question if there are a lot of negative reactions or you all think, it doesn't fit community standards.

Suppose we have a finite subset of $\mathbb N$, given by $\mathbb A=\{a_i|i\in \mathbb N_k\}$ where $\mathbb N_k=\{1,2,\dots ,k\}$. We will call $\mathbb A$ a prime set if $\exists j\in \mathbb N_k$ such that $a_j$ is prime. We will call $\mathbb A$ a composite set if it is not prime.

a. What are (if any) the tests to check the primality of $\mathbb A$? If there are no such tests, then what is the minimum knowledge that we need about the properties of the $a_i$'s or of $\mathbb A$ to comment something about the primality of $\mathbb A$?

b. Is there anything useful that we can infer if we know that $\mathbb A$ is prime?

c. Is there anything useful that we can infer if we know that $\mathbb A$ is composite?

In other words, if we have a finite set of natural numbers, is it possible to check whether we have at least one prime number among them? That is, do we have any tests or tricks to know if we have got one prime number among a few finite numbers? if not, then what more information do we need to perform such a check?

I tried to look at the product $$\prod_{m=1}^k a_m$$ but that doesn't look promising since no matter how many of the $a_i$'s are prime, this product will always be composite. Even the prime divisors of this product are of very little help since they can arranged arbitrarily in any way.

I can't think of anything more. If you all have any ideas or if you know of any work even remotely similar to this topic, please share it with me.

Please note that what I am mostly interested in is how to get some information about the individual $a_i$'s if we know some properties about the set $\mathbb A$. In other words, we don't have the privilage of peeping into the individual elements of $\mathbb A$. All we are allowed to know of, is the collection $\mathbb A=\{a_i|i\in \mathbb N_k\}$ as a whole.

Also, note that this collection is almost random. It has an upper bound and a lower bound, but it may not contain all natural numbers in that interval. So, tricks like "there must always be a prime between $f(n)$ and $g(n)$" doesn't work.

Sayan Dutta
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    If $d(n)$ is the number of divisors of $n$ then $\prod_i (d(a_i)-2) =0 \iff$ one or more of the $a_i$ is prime. This may not be very useful. – Henry Jul 12 '21 at 07:45
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    You can always check each number for primality and stop at first prime, obviously... – Sil Jul 12 '21 at 07:47
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    Basically you answered your own question. In general, there is no better way than the obvious way. In particular, large numbers are never "obviously" prime, the primality must be either already known or being established with a primality test. In some cases, we will be able to conclude efficiently that no entry can be a prime number because small factors are forced. – Peter Jul 12 '21 at 13:28
  • @Henry Yes, that is an answer, but that's not what I want. I wanted. I want to have global information about $\mathbb A$ which will lead me to the local information of the individual $a_i$'s. – Sayan Dutta Jul 13 '21 at 06:10
  • @Sil I want something more than these trivial answers. – Sayan Dutta Jul 13 '21 at 06:11
  • @Peter What I want is some tricks that we can use on the properties of $\mathbb A$ and find out some information about the properties of the $a_i$'s. – Sayan Dutta Jul 13 '21 at 06:13
  • @SayanDutta I think you can try and check if $\exists n \in \mathbb{N}_k : \nexists a \in \mathbb{N}, a\leq \sqrt{n} : a \mid n$. – Spectre Jul 13 '21 at 12:40
  • Not sure there are many tricks that will be useful ... – Roddy MacPhee Jul 14 '21 at 16:50
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    Check something called sublinear algorithms. Prof. Ronitt Rubinfeld does extensive research on this. You can check out her talks! – Pole_Star Mar 31 '23 at 09:26

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