Determining whether a Language is Turing Complete is very important when designing a language. It is a also a pretty difficult task for a lot of esoteric programming languages to begin with, but lets kick it up a notch. Lets make some programming languages that are so hard to prove Turing Complete that even the best mathematicians in the world will fail to prove them either way. Your task is to devise and implement a language whose Turing Completeness relies on a major unsolved problem in Mathematics.
The problem you choose must have be posed at least 10 years ago and must be unsolved, as of the posting of this question. It can be any provable conjecture in mathematics not just one of the ones listed on the Wikipedia page.
You must provide a specification of the language and an implementation in an existing language.
The programming language must be Turing complete if and only if the conjecture holds. (or if and only if the conjecture doesn't hold)
You must include a proof as to why it would be Turing complete or incomplete based on the chosen conjecture. You may assume access to unbounded memory when running the interpreter or compiled program.
Since we are concerned with Turing Completeness I/O is not required, however the goal is to make the most interesting language so it might help.
This is a popularity-contest so the answer with the most votes will win.
What should a good answer do? Here are some things to look for when voting but are not technically required
It should not be a simple patch of an existing language. Changing an existing language to fit the specifications is fine but patching on a condition is discouraged because it is boring. As said by ais523 in the Nineteeth Byte:
It should be interesting as a standalone esoteric language.