work on the transcendence/irrationality of (^n e) - JmsNxn - 05/17/2013
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   = 0\,\,\Leftrightarrow\,\, a_i = 0) 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:
 = 0) then we can create an isomorphism to the ring ) where  = \sum_i a_i X^i) by inventing a pointwise multiplication that is compatible with scalar multiplication of integers and has the rule  \times (^m e) = (^{n+m} e)) . However I haven't gotten very far in finding a contradiction.
I also tried using calculus and talking instead about  = \sum_i a_i (^i e)^s) 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 ^s) 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.
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