I don’t think you could refute it. I believe you could construct a binary polynomial function that gives the correct answer to every example.
For example it is difficult to reconcile the cases of 3, 12, and 19 using a reasonable-looking function, but you could solve all three cases by defining E E as the left-associative binary operation
If you give it the up-front caveat “this can represent all rational numbers and at least some algebraic irrationals”, I think that rules out the polynomial appromixation approach, since you can’t give arbitrary arguments and get intermediate values by continuity. But I’m not certain of that.
I don’t think you could refute it. I believe you could construct a binary polynomial function that gives the correct answer to every example.
For example it is difficult to reconcile the cases of 3, 12, and 19 using a reasonable-looking function, but you could solve all three cases by defining
E Eas the left-associative binary operationIf you give it the up-front caveat “this can represent all rational numbers and at least some algebraic irrationals”, I think that rules out the polynomial appromixation approach, since you can’t give arbitrary arguments and get intermediate values by continuity. But I’m not certain of that.