# Proving that $e^\pi-\pi<20$ Without Using a Calculator

No, I did not mistake the meta for the main site. :-) I'm just curious how you would recommend to approach such utterly intractable questions. Apparently, simply answering them in the negative is not accepted. The reason invoked is that one cannot simply make a statement without offering some proof: which is mainly a true and sound advice, however, the trouble with such questions lies precisely the fact that one cannot actually prove their negative. Does the site already have a policy in place for dealing with such absurd situations ?

• Merely leaving a comment leaves the question open, thus stuffing the Unanswered Questions queue. Why do I care about the Unanswered Questions queue ? Because in the very first few months of active use of the site, I had to sift through about $100$ pages of such questions, only from the tags that I follow.

• Closing it will automatically trigger the opposite reaction from many, since such questions concerning mathematical coincidences are interesting to many people, and a vicious never-ending cycle of opening-and-closing-then-reopening soon ensues.

Any other ideas ? :-\

• Your answer in particular starts with "If...". In this case, it is rather better to ask the OP if they intend to include "hand calculations" in the term "calculator"; and in such a case, then answer (or not). The starting words make your answer more into a comment (in some people's eye) and hence such action was taken. – Pedro Tamaroff Jan 9 '15 at 17:21
• Bit of an aside, but isn't the particular "question" in the title reasonably tractable, even if tedious? There exist numerous methods to bound $\pi$. Then use the Taylor series for $\exp$, and bound all but the first however many terms by some geometric series. – epimorphic Jan 9 '15 at 18:23
• @epimorphic: Answers using such an approach are usually downvoted. – Lucian Jan 9 '15 at 18:46
• Do you know of examples of such incidents? – epimorphic Jan 9 '15 at 18:48
• I remember one happening recently, but I can't seem to locate it. – Lucian Jan 9 '15 at 19:02
• I rather disagree with the decision to delete your answer, and if I had 20k rep would vote to undelete. – user7530 Jan 9 '15 at 19:44
• So, you think the question should be off-topic, but are willing to let it stay as long as it stays off the Unanswered list? – Aryabhata Jan 9 '15 at 21:14
• @epimorphic There is a, perhaps under-specified, expectation of a "readable" proof. Sure, everything that an electronic calculator does can also be done by a human calculator performing the same arithmetical operations. So it's not surprising that an answer that is a numerical method in disguise would be downvoted. On the other hand, if the requirement "without a calculator" is never made precise, the question itself can be closed as unclear. – user147263 Jan 9 '15 at 21:20
• @Aryabhata: That is the main concern, yes. That, and someone taking the time to explain to the users posting the questions why it is not reasonable to expect an elegant solution: And I refer here strictly to those that belong in the realm of sheer coincidence, not about those that are completely legitimate and fully justified. – Lucian Jan 9 '15 at 21:39
• @Lucian: I have to agree with achille hui. Unless we see such questions to be a problem, we should allow them. Yes, it is probably likely to be a coincidence (especially if the context is "I was playing with a calculator"), so what? For instance, what if I had asked, Is there a good proof $22/7$ is closer to $\pi$ than $3.14$? Would you close the question? – Aryabhata Jan 9 '15 at 21:51
• @Aryabhata: I didn't vote to close the question that you are referring to. Also, the one in your example seems reasonable. Others, however, aren't. – Lucian Jan 9 '15 at 22:02
• @Lucian: What is reasonable is entirely subjective, and the reason you see the close-reopen wars. One could of course, argue that the whole question is subjective ("not tedious", "elegant" etc). If you look at the past history of such questions, you will see that they are actually quite welcome on MSE (highly upvoted). There aren't that many, and some of them have interesting (IMO) answers. – Aryabhata Jan 9 '15 at 22:47
• It's not absurd. It's just xkcd. – John Jan 15 '15 at 2:45
• One fool can ask more questions than seven wise men can answer. (Proverb.) – Myself Jan 15 '15 at 22:16
• While these sorts of questions have some ambiguity (what exactly does 'without a calculator' mean), it is often the case that with some transformation/trick that the computation is entirely reasonable. – copper.hat Jan 23 '15 at 4:44

I find this sort of Question mildly interesting, susceptible of Answers backed by mathematical reasoning, and hence opportunities for learning something new.

While this doesn't guarantee all or even most such "utterly(?) intractable" problems will receive a well-supported Answer here, I believe it augurs against allowing unsupported Answers merely for a potential to cross them off the Unanswered list.

A positive answer can of course be supported by a demonstration. If one wants to argue for a negative answer, a start would be establishing how narrow is the "numerical coincidence". A difference in the 30th least significant digit is bound to be more difficult to resolve without "calculator" than (as here) one in the fifth significant digit:

$$e^\pi - \pi = 19.9990999\ldots \lt 20$$

Of course the Question invites ways to make such differences wider and more apparent. In any particular case it may be of interest to learn a technique for doing this, as well as the techniques for making a rational or other concise approximation adequate to making such close comparisons.

This situation is similar to Are questions of the form "has this ever been studied?" appropriate? I quote the top voted answer there, by David Speyer:

Questioners should understand, though, that if the true answer is "no" then the question will probably never be answered.

I don't have a problem with $1$ extra post in the Unanswered list. The Unanswered list is not a "queue" that one has to go through sequentially, it's a searchable database of questions that have not been resolved yet. There is nothing wrong with it being large as long as the posts therein indeed have not been resolved. (Saying "I don't think so" is not a resolution.)

Is there a danger that the unanswered list will be bloated with "Can one prove $A<B$ without a calculator"? At present there isn't: one question does not matter on the scale of $75000$. In the unlikely event that such questions become repetitive to the point of being bothersome, there is a self-correcting mechanism: when a pattern of questions becomes repetitive, they begin to draw downvotes. A negatively scored question without answers is automatically deleted when it's 30 days old or more.

• If the question becomes repetitive and they are sufficiently close to each other. Instead of downvotes, we always have the alternative to close them as abstract duplicate. Of course, we first need someone to answer one of these question in a sufficiently generic and meaningful manner. – achille hui Jan 9 '15 at 17:22
• Searchable indeed... But since I was interested in any non-trivial question involving integration, the search quickly turned into a browsing session. :-$)$ – Lucian Jan 9 '15 at 17:41
• Even after giving a "No, probably not possible" answer, the question might still stay on the unanswered list... – Aryabhata Jan 9 '15 at 18:38