TITLE: A conjecture regarding prime numbers
QUESTION [59 upvotes]: For $n,m  \geq 3$, define $ P_n = \{ p : p$ is a prime such that $ p\leq n$ and  $ p \nmid n \}$ . 
For example :
$P_3= \{ 2 \}$
$P_4= \{  3 \}$
$P_5= \{ 2, 3 \}$,
$P_6= \{  5 \}$ and so on.
Claim: $P_n \neq P_m$ for $m\neq n$.
While working on prime numbers I formulated this problem and it has eluded me for a while so I decided to post it here. I am not sure if this is an open problem or solved one. I couldn't find anything that looks like it.
My attempts haven't come to fruition though I have been trying to prove it for a while. If $m$ and $n$ are different primes then it's clear. If $m \geq 2n$, I think we can find a prime in between so that case is also taken care of. My opinion is that it eventually boils down to proving this statement for integers that share the same prime factors. My coding is kind of rusty so would appreciate anybody checking if there is a counterexample to this claim. Any ideas if this might be true or false? Thanks.
PS: I asked this question on mathstackexchage and somebody recommended I post it here as well. Here is the link to the original post  
https://math.stackexchange.com/questions/2536176/a-conjecture-regarding-prime-numbers

REPLY [9 votes]: I am working on a related project involving Grimm's conjecture.  The hope is to show that every interval of consecutive composite numbers below $10^{12}$ contains an injective divisor map, see On comparing two almost injective divisor maps for more detail.  The upshot is that there are about 700 opportunities for your event to happen (because the map $L(m)$ being largest prime factor of m is often injective, and in your case it won't be) below 2.5 times $10^{10}$, and that your event won't happen because the numbers involved are too close. (Specifically, $L(m)=L(n)=p$, and $m-n =kp$ where $L(k)$ is less than $p$ and usually less than 3, and in those cases $m/p$ and $n/p$ have sufficiently different sets of prime factors.). If I can achieve my aims while offloading data regarding your claim (e.g. a data file of the estimated 3000 $L$ pairs below $10^{12}$), I will do so and report back.  If you have several months of computer cycles to spare, I can provide a program so that you can join in the fun, AND get some data on your problem.
Gerhard "Another Opportunity For Communal Computing" Paseman, 2017.11.26.