I can tell you that typing it into google brought me a Fed Ex tracking page so it sits nicely into their tracking number system, but it’s not a real tracking number. Not that I thought it was. Just amused me that google took me straight to Fed Ex.
Then I thought each number might correspond to a letter of the alphabet, but that gives you ecgdaajjbbjehai. ( I placed 0 after 9 for this. if you place 0 before 1 then it reads fdhebbaaccafibj )
So that’s two more things ruled out. and two steps closer to figuring it out if you’re a glas half full sort of person. 
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I notice that Artem put the numbers up just before they all went out, so he can drive us nuts and we can’t beg him for the answer.
Surely he wouldn’t give us an impossible code to work out.
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Maybe you have to use the cyrillic alphabet? 
I had already considered that but even THAT doesn’t really work in THIS case.
He’s written everything else in English. No reason to expect Cyrillic.
Yes there is, if he is simply trying to be extra cryptic but as I said in my reply to Jabbath, in this case, that just doesn’t work.
I’ve just looked back and there is a 6 at the end of the number that can’t be seen on the camera so the actual number is
5374110022058196
There are 16 numbers not 15. Not that it helps me at all to work it out.
He’s just trying to remember the router password.
Is it supposed to be a date?
It would be one incredibly long date, written down with all those numbers, don’t you think?! 
He’s a very clever young man is Artem. He’s got us all still talking about him even when his apartment is empty.
Although Artem may be very clever, I’m feeling very stupid because I have no idea where to start working out what it means. 
He’s going to have to come back and tell us at some point, what it means I think.
well it’s not just a design for numbers when people want tattoos, because there are repeat numbers. you’d just have one of each if it was for design purposes.
Totally right. So no idea at all…
It’s got me baffled, that’s for sure. 
I really do think he is sat in a coffee shop at the moment laughing his head off reading these comments.
Well the sum of all digits is 56 → 11 → 2
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I don’t even know what THAT means, quite apart from the original string of numbers Artem gave us.
You sum up all the digits until you get only one single digit…
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If it was just a string of numbers to add together, putting zeros in would be a bit pointless.