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This small corner contains a minor unpublished paper of mine, New Proofs of the Irrationality of e2 and e4, which I submitted to the MAA Monthly back in 2002.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). I wrote the above paper after I had read the 2nd edition of Aigner and 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.
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)... . I couldn't be bothered to point out that the referee was ignorant of Siegel's proof of the irrationality of e2. Would the referee have said that Siegel was ignorant of Liouville's proof? And who was Siegel? Well he was a seriously great mathematician: Carl Ludwig Siegel.
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Contact details. jbcosgrave at gmail.com
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