If Wikimedia had invented its "multi-login" setup before I had created this account, I would not be in this predicament, but alas, it was not to be. I am known under two (2) usernames. At the risk of being accused as a sock-puppeteer, I make this known here. The reason is that this username is taken on a few sister sites, so I use an alternate name there, but once the multi-login came about, My alternate obtained similarly-named accounts here and elsewhere. As a result, I now contribute using both users, as whim strikes me, but mostly my new one, since it does not have any name clashes. So look nothere,butherefor my recent contributions.
Proof: Suppose,whereandare integers. Consider the
functionsdefined onby
Clearlyfor all.Letdenote the-th derivative of.Note that
;otherwise some integer
By repeatedly applying integration by parts, the definite integrals of
the functionscan be seen to have integer values. But
are strictly positive, except for the two points 0 and
,and these functions are bounded above byfor all
sufficiently large n. Thus for a large value of n, the definite
integral ofis some value strictly between 0 and 1, a
contradiction.
![{\displaystyle [0,\pi ]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb175ba2045ce5c320e0942973d0168dee16551a)
