Well, in such a case you can use the method to test somewhat more complex conjectures.
Unfortunately, the growth of the Busy Beaver function is so rapid that the number of bits of complexity of such provable conjectures added by your revised definition of N (I am ignoring the word “larger”, since that makes N undefined), is likely not very many.
What if N is larger than the number you get when you plug graham’s as both of the inputs to the ackerman function?
Well, in such a case you can use the method to test somewhat more complex conjectures.
Unfortunately, the growth of the Busy Beaver function is so rapid that the number of bits of complexity of such provable conjectures added by your revised definition of N (I am ignoring the word “larger”, since that makes N undefined), is likely not very many.