This small corner contains a minor unpublished paper of mine, New Proofs of the Irrationality of e2 and e4, which I submitted to the MAA Monthly back in 2002. A referee (see added note - of more than twenty years later - below) pointed out that the main idea was to be found in Liouville, a point evidently lost on no less a person than C. L. Siegel (see his classic book on transcendental numbers, a book that was one of my paper's references). (Originally I had written: "I also prepared an accompanying Maple worksheet", but I have decided to remove it for two reasons: only readers who have the software Maple could open it, and - while I could convert that worksheet to pdf format (so that anyone could read it), the file extension ends up being Ziegler's Proofs from the BOOK. I sent my proof to Günter Ziegler (before I heard from the Monthly Editor), and he liked it, to the extent that he was going to include it in the 3rd edition of their book.
After I heard from the Monthly, I informed Günter Ziegler, who included Liouville's argument in the 3rd edition, thanking me for bringing it to their attention. While I was disappointed to have been anticipated by Liouville, I was more pleased to receive an autographed copy of the third edition from Günter.
Some comments of late Nov. 2023. Doing some website editing, I have looked at this page for the first time since Günter kindly sent me his autographed book. I was tempted to just let the page stand as it was (here is how the page appeared in 2019, for which I have to thank the remarkable WaybackMachine), but - simply to get it off my chest, for I do not expect any readers at this stage - I want to make some comments.
One. The proofs offered in my paper were ones that I made at a time when I worked in the Mathematics Department of Manchester University, when I included them - as no more than a small aside - in an advanced course that I gave to faculty on transcendental numbers. And, while I certainly thought my proofs to be original, my personal standards were such that I did not consider they would be worthy of publication, and thus I didn't write them up for possible acceptance anywhere.
Two. Nevertheless, when shortly afterwards I attended the 1974 British Mathematical Colloquium in Sussex I decided to present them in a 15-minute talk at the Number Theory Splinter Group. My entirely undemanding talk was well-received (perhaps because everyone would have followed everything!), and I even received thank-yous afterwards from no less personages than David Masser (then a rising star) and the supernova Jean-Pierre Serre (he who could have tossed off such proofs when he was only five years old). What did their behaviour represent? Simply this: good manners.
Three. It was only in 2002 - on reading the remark by Aigner and Ziegler (that no elementary proof - along the lines of the universally known proof of the irrationality of e - that no such proof, according to them, had been given for the known irrationality of e2) in the second edition of their book, that I decided to write up what I'd had since 1972-74, and submit it to the obvious place: the Monthly. In the meantime - while awaiting a response from the Monthly I sent my proofs to A+Z. (Z. warmly responded).
Four. In the course of time an offensive/supercilious referee's report was sent to me by the then Editor of the Monthly - the Editor wasn't offensive, it was the (as per practice) anonymous referee. I didn't keep it I should add, but I very much remember its content/tone, and it went something like this: the author is clearly ignorant of the history. All he had to do was go into his library and there find Liouville's proof (dating from the mid 1800s)... .
Five. What could a well-mannered, sensitive referee have written as a report? I would suggest something along these lines:
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