work on the transcendence/irrationality of (^n e)

[JmsNxn]

Hey everyone, I'm wondering if anyone knows of any work done on proving the irrationality of or perhaps transcendence. It seems like a fruitful question in my eyes, and I think that no doubt these constants are probably transcendental. A proof of this would be quite spectacular though, and I imagine it would require some ingenious argument. Irrationality maybe a bit more modest, and I expect easier to prove.

I was wondering because I was trying to prove something similar but slightly stronger, that if which I think is perfectly reasonable. I was trying to approach this using ring theory, saying that by contradiction we have some relation and it is the smallest such one, so:

then we can create an isomorphism to the ring where by inventing a pointwise multiplication that is compatible with scalar multiplication of integers and has the rule . However I haven't gotten very far in finding a contradiction.

I also tried using calculus and talking instead about and learning about where the zeroes of such functions are distributed.I think this is probably the more fruitful method. I think it is very unlikely that the zeroes of a function like are algebraic. I was going to try and go by induction, since when the biggest term is the zeroes are transcendental then assume when the biggest term is the zeroes are transcendental and go from there.

Any tips or hints or knowledge would be greatly appreciated. I think proving is transcendental or irrational is a big step in investigating tetration.