 work on the transcendence/irrationality of (^n e) - Printable Version +- Tetration Forum (https://tetrationforum.org) +-- Forum: Tetration and Related Topics (https://tetrationforum.org/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://tetrationforum.org/forumdisplay.php?fid=3) +--- Thread: work on the transcendence/irrationality of (^n e) (/showthread.php?tid=793) 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 $$^n e$$ 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 $$a_i \in \mathbb{Z}$$ $$\sum_i a_i (^i e) = 0\,\,\Leftrightarrow\,\, a_i = 0$$ 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: $$\sum_i a_i (^i e) = 0$$ then we can create an isomorphism to the ring $$\mathbb{Z} [X] / p(X)$$ where $$p(X) = \sum_i a_i X^i$$ by inventing a pointwise multiplication that is compatible with scalar multiplication of integers and has the rule $$(^n e) \times (^m e) = (^{n+m} e)$$. However I haven't gotten very far in finding a contradiction. I also tried using calculus and talking instead about $$f(s) = \sum_i a_i (^i e)^s$$ 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 $$f(s)$$ are algebraic. I was going to try and go by induction, since when the biggest term is $$e^s$$ the zeroes are transcendental then assume when the biggest term is $$(^{n-1} e)^s$$ the zeroes are transcendental and go from there. Any tips or hints or knowledge would be greatly appreciated. I think proving $$(^n e)$$ is transcendental or irrational is a big step in investigating tetration.