I have written a program that can check them.

As for proving a counter example or proving there are none, that's rather difficult I think.

The only way I can think of showing that a solution is true is by an exhaustive algorithm. If someone were to say they have a solution, I can verify it computationally using a program that exhausts all possible parsings of the numbers and symbols, and shows none of them equal the target number.

Whether or not you see this as a proof is your call, you'd have to have confidence that my program really did check all possibilities.