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I have the proof for $\pi + e$ being irrational.

Honestly, I’ve been delving into irrational analysis for awhile now, and I got lucky stumbling into a solution. I’m not an amateur mathematician, but credentially-bottom. Would love to share because I know its interesting, but is this a good place to offer an argument?

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    $\begingroup$ Asking whether an argument you have is correct may be a reasonable question to post here. $\endgroup$
    – Michael Hardy
    Mar 20, 2021 at 7:19
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    $\begingroup$ Or you could upload a preprint to arxiv(dot)org. $\qquad$ $\endgroup$
    – Michael Hardy
    Mar 20, 2021 at 7:19
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    $\begingroup$ Let's see your proof, you can answer the question $\endgroup$
    – Aderinsola Joshua
    Mar 20, 2021 at 7:20
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    $\begingroup$ IMO, this is not really a good place for that. Have a math instructor/professor you know talk with you about it; maybe you're right and can get it published. But to be honest a lot of people come to MSE with posts in the vein of "I have proved/disproved [longstanding open conjecture/problem]," and it gets a bit tiresome. (A lot of the time these often tend to be the same copy-and-paste proof someone found online, or they are often severely lacking in one or more respects.) (cont.) $\endgroup$
    – PrincessEev
    Mar 20, 2021 at 7:23
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    $\begingroup$ Or at least, that's usually when you as an amateur are assuming you're right to begin with. A healthy level of skepticism and tempering of one's expectations should be encouraged. Now, if you want people to look over your alleged proof for possible flaws (rather than just to share your alleged proof), MSE will be open to that -- just be sure to take the criticism in stride if and when it comes, rather than stalwartly assuming you're right. $\endgroup$
    – PrincessEev
    Mar 20, 2021 at 7:23
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    $\begingroup$ Thank you everyone here for all the kind help. I’ll see what steps I can take after typing it up in LaTeX before wonderful skeptics such as you all can have a chance to reason against it. $\endgroup$
    – Ace Sarich
    Mar 20, 2021 at 17:46
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    $\begingroup$ If its multiple pages long then you are probably better off putting it in arxiv and/or sending to a journal IMO $\endgroup$
    – Calvin Khor
    Mar 21, 2021 at 1:56
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    $\begingroup$ There is a relevant question on main here, about what to do with a proof of an open problem. $\endgroup$
    – user1729
    Mar 21, 2021 at 8:42
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    $\begingroup$ You should post it here. Some people with grumble but plenty of people will enjoy taking a look. Eventually if it's not worthy of attention, it will be downvoted and fall out of sight which is no inconvenience to anybody. $\endgroup$
    – it's a hire car baby
    Mar 21, 2021 at 9:52
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    $\begingroup$ @TeresaLisbon the link you put in one of your comments doesn't work, the page has been deleted. $\endgroup$
    – A-Level Student
    Mar 21, 2021 at 15:54
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    $\begingroup$ @A-LevelStudent You should probably be aware that some real big names in math frequent the website. It's not like they all sit on MO, I know some real big shots here. You may call yourself an A-Level student, but you have an account on MSE, so in no time you will be a super star student! I've had conversations with people here that I could never have thought I'd meet in real life. $\endgroup$
    – Sarvesh Ravichandran Iyer
    Mar 21, 2021 at 16:31
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    $\begingroup$ @A-LevelStudent: publishing any content on any public forum will ensure that any actions of plagiarism can be handled if it comes to notice. This is simply because users in general don't have access to modify time when a content was published. Don't worry about it. $\endgroup$
    – Paramanand Singh Mod
    Mar 22, 2021 at 1:58
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    $\begingroup$ @Aruralreader $\lambda = \frac{-e}{\pi}$? But you can try the following simpler exercise : either $\pi + e$ is irrational or $\pi e$ is irrational (hint : Vieta's formula) and see if you can extend this logic to $\pi + \lambda e$ for $\lambda$ rational. It gives an either/or statement which is very useful. $\endgroup$
    – Sarvesh Ravichandran Iyer
    Mar 24, 2021 at 4:56
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    $\begingroup$ @TeresaLisbon from your hint, I've thought of a possible proof: consider the expression $(x-e)(x-\pi)$. Since $e$ and $\pi$ are both transcendental, at least some of the coefficients of this polynomial must be irrational; hence at least one of $\pi+e$ and $\pi e$ is irrational. Is that right? $\endgroup$
    – A-Level Student
    Mar 24, 2021 at 20:28
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    $\begingroup$ If you do end up publishing it in another medium, please leave a link here. A lot of us are curious to see what the proof consists of. $\endgroup$
    – user400188
    Mar 26, 2021 at 0:08

2 Answers 2

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$\begingroup$

No, this is not the correct place to post your solution. At best this looks like self-righteous advertising, at worst spam.

There are a few caveats and points for discussion here though.

  1. This is a Q&A site. Saying "here is my proof" is not a question...
  2. The problem of determining if $\pi+e$ is irrational is stated on Wikipedia as being open, and so you should read the question What should an amateur do with a proof of an open problem? .
  3. If the problem you solved were not Wikipedia-famous then my answer might be different.
  4. An acceptable question might be something like: "I was thinking about this problem and got lucky stumbling into what looks like a solution. I cannot find an error in it. However, as the proof is so short and elementary I was wondering if any of you could find the error? Thanks." This is a question (so addresses (1)), and shows that you are sceptical of your result*.

*Half of maths is being sceptical, including of your own proofs and also of those in papers by fancy people published in fancy journals.

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    $\begingroup$ You should add that people in chatrooms might be interested to discuss this, and I don't think they will ask to be very specific. In fact, the mathematics chatroom will be able to reply very quickly to the proof, I'd think! $\endgroup$
    – Sarvesh Ravichandran Iyer
    Mar 22, 2021 at 14:35
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    $\begingroup$ Objecting on the basis that an already-written proof is not a question is silly. There is an ‘answer your own question’ checkbox when asking a new question, and it’s pretty easy to phrase the problem in the form of a separate question and answer. Now, whether the proof is correct is another matter… $\endgroup$
    – user3840170
    Mar 24, 2021 at 9:02
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    $\begingroup$ @user3840170 So then the "question" would be the famous problem, and the "answer" would be a(n attempted) proof? That doesn't make a lot of sense, since a famous problem would make a poor question post, and an the attempted proof posted as an answer doesn't really facilitate trying to get feedback on the proof. $\endgroup$
    – xxxxxxxxx
    Mar 24, 2021 at 9:23
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    $\begingroup$ @user3840170 That point is the least of my objections, and can be easily overcome (e.g. by phrasing it like in point 4). My main point, however, is point 2 (and none of the answers in the linked question say "post it here!"). $\endgroup$
    – user1729
    Mar 24, 2021 at 11:40
  • $\begingroup$ @TeresaLisbon I think posting to chat depends on what they are hoping to get out of it. If they are just hoping for some informal feedback then chat might be OK. I think anyone posting such a solution anywhere on the internet should be thinking "Why am I posting this here? What am I hoping to get out of this?". A lot of the answers to the question I linked to suggest posting on the arXiv, but why? What's the point? It's not a magical place where arXiv papers are automatically accepted by the community. $\endgroup$
    – user1729
    Mar 24, 2021 at 11:56
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    $\begingroup$ (Also, regarding this question, I agree strongly with Did's comments there, so I feel I should highlight them.) $\endgroup$
    – user1729
    Mar 24, 2021 at 12:00
  • $\begingroup$ @user1729 In agreement with your comments, I just felt that if the user wanted to do something with this site and that proof, then chat is consultable. Of course, it is highly informal, but the people who are there are qualified enough to say something useful. $\endgroup$
    – Sarvesh Ravichandran Iyer
    Mar 24, 2021 at 12:34
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    $\begingroup$ @user1729: what's the point? Well, sharing such a marvelous piece of proof to everyone itself is the most important point. Any professional benefit including fame and money pales in comparison to that. Who knows there might be another Apery moment. $\endgroup$
    – Paramanand Singh Mod
    Mar 26, 2021 at 4:07
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    $\begingroup$ @ParamanandSingh My question is not "what is the point of sharing it?" but instead "what's the point of sharing it there?" (specifically the arXiv, but also on Math.SE or anywhere else). Anyone in such a situation should be considering the most effective way to distribute their proof. I hold opinions on this matter (posting it on the arXiv, viXra or here would be ineffective, while journal system is designed precisely for this purpose) but my point is that anyone in this situation should consider this question (and possible ignore my opinion). $\endgroup$
    – user1729
    Mar 26, 2021 at 9:40
  • $\begingroup$ @user1729: oh I understood your intent. Unless the motive is related to professional benefit, I think mathse has a wider outreach. Just share it with any of the academia persons whom you respect here (via chat) and such a thing like irrationality of $\pi+e$ will get viral in minutes. $\endgroup$
    – Paramanand Singh Mod
    Mar 26, 2021 at 9:42
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    $\begingroup$ I object to the phrasing "Half of maths is being skeptical, including of your own proofs and also of those in papers by fancy people published in fancy journals." I would say half of maths (maybe more or less, not sure the %) is verifying (which also helps in ingesting material). The difference being that "I have not verified X" or "I tried to verify X but there is a flaw in the proof" is a lot different from "I doubt X" or "I don't expect anyone outside a certain circle to prove X". Verification is objective and rigorous, skepticism is sociological and lazy. $\endgroup$
    – John_Krampf
    Mar 26, 2021 at 11:18
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    $\begingroup$ @John_Krampf saying "half of maths is verification" implies that I spend half my time verifying other peoples proofs. This is not true. My point was really about having a mental state of not believing things based on who wrote them (including if it was yourself) or where it was published. It's not saying "I doubt X" to other people, but instead being aware that mistakes are easy to make by anyone, and result should not be judged superficially. $\endgroup$
    – user1729
    Mar 26, 2021 at 11:41
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    $\begingroup$ And there is the other side of the story. When everyone was thinking that "primes is in P" was a damn difficult open problem, AKS came up with a very smart simple proof. $\endgroup$
    – Paramanand Singh Mod
    Mar 26, 2021 at 14:31
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    $\begingroup$ @user1729: Recently Lisa Hayden has attempted to prove the Riemann Hypothesis, and posted about it here. Unfortunately, despite the great amount of effort she devoted to this attempted proof, her post only served to demonstrate that attempted proofs of open problems just aren't very suitable for this site. $\endgroup$
    – Joe
    Jun 14, 2021 at 14:19
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    $\begingroup$ @Joe in the words of Andreas Blass, who commented on the double-post to MathOverflow, "Alleged proofs belong on the arXiv, not on MathOverflow." $\endgroup$
    – user1729
    Jun 14, 2021 at 16:05
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It's something that users have posted here before (an incorrect proof that $\pi + e$ is irrational).

If you have a correct proof, you could give it as a new Answer to Proof of $\pi$ + $e$ irrational. But you should do the research and check that you are not offering an incorrect proof much like the attempts that have been posted here previously.

The older post was closed, reopened, reclosed, and deleted

Apparently the exposure of that Question here on meta inspired a group of close votes a few hours ago. It had been asked and answered seven and a half years ago, so the timing suggests cause-and-effect.

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    $\begingroup$ I would hope that the OP's proof is incomparably better than the "proof" given at your link! It would be a mistake to post it there as an answer, in my opinion. $\endgroup$
    – TonyK
    Mar 21, 2021 at 12:20
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    $\begingroup$ For my numerous down voters, I'd point out that my invitation to post an Answer on the main site is conditioned on "If you have a correct proof". Although it is likely that no such proof has been stumbled upon, it is part of our mission to collect good content and help learners of mathematics at all levels. Which of us has not learned something from their mistakes? Some of which I've posted before having my confusion pointed out... $\endgroup$
    – hardmath
    Mar 22, 2021 at 15:46
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    $\begingroup$ @hardmath: your conclusion about linked post and its closure is correct. I had seen that post long back and downvoted it but had forgotten until it came up here. $\endgroup$
    – Paramanand Singh Mod
    Mar 22, 2021 at 17:11
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    $\begingroup$ @hardmath: the question you linked is about a concrete (and very naive) attempt a person made at the question. A proof would not answer the question (and the current answer is perfect). $\endgroup$
    – Martin Argerami
    Mar 23, 2021 at 4:04
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    $\begingroup$ My downvote is actually based on the fact that, if they do have a correct proof, I think it is an incredibly poor idea to post it on MSE (as an answer to the linked post or otherwise). This is an open problem and they should work on getting it written up and published. $\endgroup$
    – xxxxxxxxx
    Mar 24, 2021 at 9:26
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    $\begingroup$ The older post has been reopened now. $\endgroup$
    – Joe
    Mar 25, 2021 at 16:42
  • $\begingroup$ @Joe It is deleted... $\endgroup$
    – cqfd
    Jul 5, 2021 at 5:14
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    $\begingroup$ @ShiveringSoldier: After I posted my comment, it was closed again, then deleted. I have voted to undelete. I see nothing wrong with that question. $\endgroup$
    – Joe
    Jul 5, 2021 at 5:16
  • $\begingroup$ I agree. I cast my vote a few days ago, and I was surprised to find out that it remains deleted. $\endgroup$
    – cqfd
    Jul 5, 2021 at 5:20
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    $\begingroup$ @ShiveringSoldier: It's sad that this meta thread has generated such controversy surrounding that question. $\endgroup$
    – Joe
    Jul 5, 2021 at 5:32

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