Let $n$ be a positive integer , $p_k$ the k-th prime number and $a_j=-1$ or $a_j=1$ for $j=1,\cdots ,n$
What is the minimum value of $$S:=|\sum_{j=1}^n \frac{a_j}{p_j}|$$ ?
Motivation : If we multiply such a number with the product of the first $n$ primes (which is $p_n$# , let us call it $P$) , then $S\cdot P$ is divisible by no prime factor upto the n-th prime number since every summand except one is divisible by a particular prime number among the first $n$ primes. Therefore, if we can make $S\cdot P$ small , the chance for a prime number is good.
Aditionally : Can we prove that $S\cdot P=1$ is impossible for $n>3$ ?
For $4\le n\le 25$ , $S\cdot P=1$ is impossible. The smallest values for $S\cdot P$ , when $1$ is excluded , are summarized in a table using PARI/GP :
gp > for(k=2,25,p=prod(j=1,k,prime(j));mini=10^1000;forvec(z=vector(k,j,[0,1]),a=vector(length(z),j,2*z[j]-1);s=abs(sum(j=1,length(z),a[j]/prime(j)));n=s*p;if(n>1,if(n<mini,mini=n;merk1=a;merk2=s*p)));print(k," ",merk1," ",merk2," ",isprime(merk2,3)))
2 [-1, -1] 5 1
3 [-1, 1, -1] 11 1
4 [-1, 1, 1, -1] 23 1
5 [-1, 1, 1, -1, 1] 43 1
6 [-1, 1, 1, 1, -1, -1] 251 1
7 [-1, 1, 1, -1, 1, 1, -1] 263 1
8 [-1, 1, 1, 1, -1, -1, -1, 1] 21013 1
9 [-1, 1, -1, 1, 1, 1, 1, -1, 1] 1407079 0
10 [-1, 1, 1, 1, 1, -1, -1, -1, -1, -1] 4919311 0
11 [-1, 1, -1, 1, 1, 1, 1, -1, 1, 1, -1] 818778281 0
12 [-1, 1, 1, 1, -1, -1, 1, -1, -1, 1, -1, 1] 2402234557 0
13 [-1, 1, -1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1] 379757743297 0
14 [-1, 1, 1, -1, 1, -1, 1, -1, 1, 1, 1, 1, -1, -1] 3325743954311 0
15 [-1, 1, 1, -1, -1, 1, 1, 1, -1, 1, -1, 1, 1, 1, -1] 54237719914087 0
16 [-1, 1, -1, 1, 1, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, -1] 903944329576111 0
17 [-1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, 1, 1, 1, 1, -1, 1] 46919460458733911 0
18 [-1, 1, 1, -1, 1, 1, -1, 1, 1, -1, -1, -1, 1, -1, -1, 1, -1, 1] 367421942920402841 0
19 [-1, 1, 1, 1, -1, -1, 1, -1, 1, 1, -1, 1, 1, -1, -1, -1, -1, -1, -1] 17148430651130576323 0
20 [-1, 1, 1, -1, 1, -1, -1, 1, 1, -1, 1, -1, 1, 1, 1, 1, 1, -1, -1, 1] 1236225057834436760243 0
21 [-1, 1, 1, 1, 1, -1, -1, -1, -1, 1, -1, -1, 1, -1, 1, -1, 1, -1, -1, 1, -1] 4190310920096832376289 0
22 [-1, 1, 1, 1, -1, -1, -1, -1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1] 535482916756698482410061 0
23 [-1, 1, 1, 1, -1, -1, -1, 1, -1, 1, 1, -1, 1, 1, -1, -1, 1, 1, -1, 1, -1, -1, -1] 29119155169912957197310753 0
24 [-1, 1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1, 1, -1, -1, -1, 1, -1, -1, 1, -1, -1, 1, -1] 443284248908491516288671253 0
25 [-1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, 1, -1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1] 28438781483496930396689638231 0
gp >
Unfortunately, for $9\le n\le 25$, the minimum value is not prime. Perhaps, the minimum value still grows too fast with $n$.
An extension of the table or an estimate of the minimum value and the chance to find a prime would be very appreciated.