The Rationality of for Non-Integer Rationals

This is an interesting question in transcendental number theory. Let me analyze it step by step.

What We Know

First, let's clarify what's already established:

1. For (rational but not an integer), we know that is irrational.

2. Using the Gelfond-Schneider theorem, we can prove that is transcendental (and hence irrational).

Analyzing

For the next level, , we have:

• Base: (rational)
• Exponent: (transcendental)

This presents a specific form: a rational number raised to a transcendental power.

Here's where we reach a limitation. The Gelfond-Schneider theorem addresses algebraic bases raised to algebraic irrational powers, but not algebraic bases raised to transcendental powers.

Current State of Knowledge

To determine whether can be rational, we would need a theorem addressing what happens when we exponentiate a rational number to a transcendental power.

While it seems intuitively likely that is transcendental (and thus irrational), I'm not aware of a proven general result that would resolve this question definitively.

Schanuel's conjecture, if proven, would imply that is indeed transcendental, but this conjecture remains open.

Conclusion

The question of whether can be rational for some appears to be an open problem in mathematics. The existing tools in transcendental number theory don't seem sufficient to provide a definitive answer.

I agree with your intuition that this is likely an open problem in the literature. If anyone knows of specific results addressing this question, they would be valuable contributions to this discussion.