Denote the set of prime numbers by $P$.
Let $u,v,m,n \in \mathbb{N}-\{0\}$ satisfy:
$m \leq n$, $\gcd(u,m)=1$ and $\gcd(v,n)=1$.



> 
> Is it possible to find $L \in \mathbb{N}$, such that $u+Lm \in P$
> and $v+Ln \in P$? (different primes, probably).
> 
> 
> 


Of course, by [Dirichlet's theorem on arithmetic progressions](https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions), we can find $L$ such that one of $\{u+Lm,v+Ln\}$ is prime. But are those $L$'s 'dense enough' to guarantee that both $\{u+Lm,v+Ln\}$ are primes?


A similar, hopefully a less difficult question, is as follows:



> 
> Is it possible to find $L \in \mathbb{N}$, such that:
> 
> 
> (i) $u+Lm < v+Ln$;
> 
> 
> (ii) $u+Lm = p \in P$;
> 
> 
> (iii) $p$ does not divide $v+Ln$?
> 
> 
> 


From the answer to [this](https://math.stackexchange.com/questions/2811792/specific-u-v-m-n-in-mathbbn-for-which-there-exists-l-in-mathbbn-suc) question, it is clear that there exists $L \in \mathbb{N}$ such that:
(i) $u+Lm < v+Ln$ and (ii) $u+Lm = p \in P$. The problem is how to guarantee that $p$ will not divide $v+Ln$?


Thank you very much!


**A remark about the answer:** Let us concentrate on the special case mentioned in the answer: $m=n=1$, $u=1$, $v=3$, so $A:=1+L$ and $B:=3+L$. There is a major difference between asking: 'Are there infinitely many $L$'s such that $A,B \in P$' (very very difficult question) and 'Does there exist $L$ such that $A,B \in P$',
which is a very easy question that has a positive answer, for example $L=2$ yields $(A,B)=(3,5) \in P^2$. Therefore, I expect that my (first) question has a positive answer, or maybe I am missing something, and even finding only one such $L$ is a difficult task? (I do not require that $A$ and $B$ will be greater than a given number).


**Edit:** I have now noticed that my (first) question has already been asked before on MO, [here](https://mathoverflow.net/questions/217680/primes-in-simultaneous-arithmetic-progressions?rq=1). Now I see that my (first) question is just [Dickson's conjecture](https://en.wikipedia.org/wiki/Dickson%27s_conjecture) with $k=2$, $\gcd(a\_1,b\_1)=1$ and $\gcd(a\_2,b\_2)=1$ (in Wikipedia notations). In order to avoid 'trivial' counterexamples such as the one mentioned in the comments (namely, $3+L$ and $4+L$, with always one of the two necessarily not a prime number, since it is even), additional conditions
(["congruence condition"](https://math.stackexchange.com/questions/2930958/concerning-dicksons-conjecture/2930972?noredirect=1#comment6054677_2930972)) must be imposed.


[This](https://arxiv.org/pdf/1410.8198.pdf) is a relevant paper that I have now found (it is from 2015);
interestingly, it mentions a result of Maynard-Tao that almost answers Dickson's Conjecture in case $k=2$, $b\_1=b\_2=1$.



Almost simultaneous Dirichlet's theorem on arithmetic progressions
This is simply a reformulation of (a weaker version of) the prime pairs conjecture, a generalization of the twin primes conjecture. For example, when $m=n=1$ and $u=1$, $v=3$, this is exactly the twin primes conjecture, except that such conjectures are usually formulated as "are there infinitely many?" rather than "does there exist one?".


It would seem reasonable to think that the latter type of question is easier than the former, but in general that doesn't seem to be the case. Hence this is an open problem.


