I am not a specialist in the theory of algorithms nevertheless I hope that the following proposal is new.
Each algorithmically irresolvable problem being coded in any way yields any large natural number, which may be called "irresolvable" too. Firstly such a number was constructed by Goedel, but his number is too large.
May there exist any "natural" coding of algorithmical problems by natural numbers (by "sufficiently dense" subset of natural numbers) ?
If we choose such a coding then we may introduce obvious definitions of "resolvable" and "irresolvable" numbers. If we prove, for instance, that 1000 is (absolutely) resolvable and 1000000 is irresolvable (which fits an intuition) then it would be a general result both for the set of natural numbers and for the theory of algorithms.