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I'm a physics graduate but have written a number theory paper as a hobby. A couple of professors have read it briefly though it is not their field they gave positive feedback and suggested I send it to professors working in this area. I sent it to 8 professors and none has replied to me. I asked the original professor what's going on and he said probably the following: The nature of the paper (Fermat's last theorem) and being an amateur mathematician it will invoke memories of all the failed attempts on this problem over the centuries. Even if it is correct that doesn't matter they will throw it in the bin without reading it. He suggested I may have to pay a qualified reader to evaluate it so I'm wondering how do I find a qualified number theory reader. There are no listings for maths readers and I don't want to email everyone on the number theory web until I find one.

Thank you for any advice.

Chris S
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Well, I am a professional number theorist, so let me give you some feedback that you don't seem to have received yet:

Your paper containing an elementary proof of Fermat's Last Theorem is wrong.

This is what I think, and I am confident that this is what more than 99 out of 100 professional number theorists would think. Here by "wrong" I mean as a real person would use the term: i.e., probably wrong. I obviously don't know that it is wrong, and I couldn't without reading it, but in real life there is almost nothing that one knows is right or wrong: we have to go based on probability. Let me try an analogy: suppose I am an amateur aerospace engineer. I claim that I can walk into anyone's home and, using only the materials that are already there, build a ship that can safely carry me to the moon and back. Wouldn't you say that I'm wrong? Wouldn't you say that, if anything, the fact that we have been to the moon and back gives us a better sense of the resources necessary for that, and that gives you more confidence that my estimation is wildly off? That while it's of course possible that I'm right, if I were I would have to be in possession of such vastly superior knowledge and expertise that I could easily provide evidence of it on a smaller scale? That's pretty much how we number theorists feel.

Let me make one point about amateur mathematicians: it is exceedingly rare for an amateur mathematician to solve a problem that has been worked on unsuccessfully by the mathematical mainstream, but it has happened sometimes (more or less depending upon what you mean by an amateur mathematician, but if for instance you count brilliant young people without a college degree: sure, it has happened several times). However I know of no examples in which an amateur mathematician has solved a significant mathematical problem without having done other substantial mathematical work already. So...is this your first math paper, or the first substantial piece of mathematics you think you've done? If so, I really think it's wrong: that's just not how success in any field works. Nobody picks up a tennis racket for the first time and discovers they can play at the professional level. On the other hand, if you do already have published papers, then certainly mention that in your correspondence: it will make a big difference.

Anyway, if you want to find out why your proof is wrong, then yes: I agree with the other answers who say that you should pay to enlist the services of a tutor. If you go to any math department with number theorists and say that you have reading and thinking about FLT and have some ideas you'd like to clarify and are willing to pay, say, $30 an hour to get one-on-one tutoring, I think you're very likely to find some takers.

Let me say finally that my goal in writing this answer is not at all to crush your dreams. Rather, I hope that you actually like mathematics enough to move past your thought that you have proven FLT and engage with the subject matter more deeply. Mathematics is a lot deeper and richer than I think you know...in some wonderful ways. Good luck.

Pete L. Clark
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    I might add that this happens to professional mathematicians as well: they stumble upon what look like solutions to big open problems. It has certainly happened to me. We too, assume that such proofs must be wrong, and savagely tear through them until we inevitably uncover a mistake. In the very unlikely event that we don't find a mistake, we might venture to show them to colleagues, in which case our firm expectation is that they will find the mistake. – Anonymous May 06 '17 at 20:16
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    For what it's worth, be careful to say FLT and not FTL. You will look more unbelievable otherwise. – sgf May 06 '17 at 21:59
  • Jeez I feel like Hippacus about to be hurled overboard by claiming that the sqr root of 2 is not a rational number. – Chris S May 07 '17 at 07:57
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    Please understand that you're the 189784678th person to claim to have proved FLT using elementary methods, and that the previous 189784677 claims were wrong, sometimes for very subtle reasons, but almost always because of an elementary mistake or utter incoherence. Your claim will be held to a higher standard of proof, because you are claiming to do something that centuries of stellar mathematicians have found impossible. That's doesn't mean you're wrong, but it does mean experts will be predisposed not to believe you. There's no point being insulted by this; it's not about you. – JeffE May 07 '17 at 13:28
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    @Chris we're all sorry that you feel like Hippasus, but logically speaking surely you realize that comparing your emotions to those of Hippasus doesn't make it any more likely that your claims are correct in the way that Hippasus's claims were, right? Pete gave you a straight, honest answer. He is a highly reputed professional mathematician, number theorist and academia.se contributor. Frankly you will not get a better quality answer to your question from anyone else anywhere. I suggest listening to the content of Pete's message instead of focusing on the emotions it makes you feel. Good luck! – Dan Romik May 07 '17 at 17:10
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    There is a difference with Hippasus. The story is that the others were shown the proof and understood that it was true but they didn't want it to be spread. Here we've only seen a claim, not the proof (correct or not). – ypercubeᵀᴹ May 07 '17 at 20:12
  • @ChrisS Pete's response is actually quite kind and considerate. You could also post your "proof" online, so that we can point out the (most likely) obvious mistakes. – Rüdiger May 08 '17 at 10:26
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    @Rüdiger The thing is that the mistakes need not be at all obvious. A common type of mistake in such proofs consists of huge leaps in logic, making the proof incomplete to the standards of the mathematical community. Authors of such proofs are usually completely incapable of accepting that there is an issue, especially since everything might be correct (after all, the claimed statement is known to be true, so the proof need not have any statements that are actually wrong, just ones that have not been proven properly). – Tobias Kildetoft May 08 '17 at 11:01
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I commend your modesty. Many people, in a similar situation as you, seem angry that the mathematical world has not recognized their genius. Your attitude is much more modest, and you seem to be willing to do whatever is needed to get your paper read.

I recommend addressing your question to a graduate student rather than a professor. At least if you are in the United States, now (early May) is a particularly good time to do so: final exams just ended, and many students are a lot less busy and would like to earn some money over the summer.

Moreover, I would frame your request as seeking out a private tutor. You can often find these by Googling. For example, here is a list of math Ph.D. students at the University of Wisconsin who have made themselves available for (paid) private tutoring. Hopefully you could arrange this with someone within driving distance of wherever you live, so that you can do this in person.

I would send a brief e-mail something like the following:

Dear XXX,

My name is ZZZ and I'm an amateur hobbyist mathematician. I believe that I have found an elementary proof of Fermat's Last Theorem, and one that covers further cases as well. Since I'm an amateur, I understand that I could have easily made a mistake -- and indeed that this might well be the most likely outcome.

I'd like to hire you to discuss my paper with me, to check my arguments, to point out any logical flaws that you see, and to offer me your advice. I'm happy to pay $60/hour and to meet with you at your convenience, and to send you my paper in advance if you like. Would you please be in touch if you're willing to do this?

Thank you very much.

Possible reactions will vary. Most people will ignore you, so be prepared to e-mail multiple people. Some might ask to see your paper; if it is logically coherent but contains a mistake, someone might point out the mistake and tell you not to worry about payment.

If you managed to hire a tutor, and if you have, in fact, come up with an elementary proof of Fermat's Last Theorem (as you claimed in the comments), and if moreover you have explained it clearly and without mistakes, then whoever you hired will likely refuse payment and bring your work to the attention of faculty members.

Good luck to you!

Community
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Anonymous
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  • Wow! Yes I will try that approach and thank you for that wise advise. – Chris S May 06 '17 at 03:33
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    I really liked your answer up until this point: "If you managed to hire a tutor, and if you have, in fact, come up with an elementary proof of Fermat's Last Theorem (as you claimed in the comments), and if moreover you have explained it clearly and without mistakes, then whoever you hired will likely refuse payment..." While the implication is probably true, the premise is exceedingly unlikely. I wouldn't encourage the OP to go in to this transaction hoping to get his money refunded at the end. That seems to be setting him up for disappointment. – Pete L. Clark May 06 '17 at 19:13
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    @PeteL.Clark: I definitely understand your point. That said, although I agree (unfortunately) that the premise is unlikely, the OP nonetheless believes it, and I am tried to write my answer with his/her point of view in mind. OP wrote in a comment: "I sometimes wish I never found [this] due the large amount of time I spent on it." If OP determines that his/her proof is wrong, then surely that will be a bigger disappointment than not getting a refund? – Anonymous May 06 '17 at 20:00
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    That said, to the OP, @PeteL.Clark is completely correct: if you follow my advice, and if a tutor agrees to work with you, you should go into it anticipating that the tutor will find a mistake. (Or, possibly, that your paper is written in such a nonstandard way that the tutor can't follow it at all, in which case h/she might give you advice on mathematical writing). To a mathematician, your claim will sound like you are claiming to be able to run a marathon in under an hour. The problem is just that difficult. – Anonymous May 06 '17 at 20:11
  • +1 for "now (early May) is a particularly good time to do so: final exams just ended, and many students are a lot less busy and would like to earn some money over the summer" Also, regarding "then whoever you hired will likely refuse payment and bring your work to the attention of faculty members", Chris S need not worry about the person stealing the proof --- he/she will be plenty famous as it is for being the person who "discovered you". – Dave L Renfro May 08 '17 at 14:15
  • Just an update, I got a response at last! I have found a reader/editor. He is the chief editor of a number theory journal which is a bonus. He's willing to review my work. He's done the first 3 hours and so far so good, but keep those torches burning - I'm still strapped to the heretic burning stake. See if you can find my paper online it's in 2 locations but once it is edited I'll put it on arXiv if it is ok with the journal. – Chris S Jun 06 '17 at 07:57
  • i would love to see it, please share a link. –  Jun 08 '17 at 13:03
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An alternate approach would be to formalize your papee in a mechanical proof system, like Coq or Agda. This has several advantages:

  • If your proof is correct, it is undeniably correct, and others will take you seriously
  • It will make the assumptions you have in your paper explicit to you, so you can possibly spot where you've made mistakes
  • If your proof contains tedious but repetitive parts they can be automated. (This is especially true of Number Theory, where induction is common, as opposed to real analysis)
  • If all else fails, you will have learned a fun, interesting new tool

These tools have helped some results, like the 4 color theorem, gain more acceptance in the community.

Joey Eremondi
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Have you thought of soliciting feedback on MathOverflow? E.g., you can post your proof on some publicly accessible location on the web and post a question on MathOverflow requesting feedback. If you offer to pay as some other people were suggesting that may increase the chance that you'll get help, but if your proof is written in clear and readable language and interesting to examine it's completely possible that knowledgeable people will offer you some useful advice free of charge. Good luck!

Dan Romik
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    I could not seem to find it while searching now, but I am fairly certain the attitude of MO is that questions asking about proofs of well-known hard open problems are off-topic. – Tobias Kildetoft May 07 '17 at 17:27
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    Please, no. Do not post on MO about this. – Andrés E. Caicedo May 07 '17 at 17:33
  • @AndrésE.Caicedo is that your personal preference or official MO policy? – Dan Romik May 07 '17 at 17:40
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    Indeed, https://meta.mathoverflow.net/a/942/4832 and https://meta.mathoverflow.net/questions/1818/questions-of-the-type-i-have-written-something-please-give-me-feedback make it clear that this is not an appropriate type of question for MO. There isn't really "official" policy, but it seems to be a clear community consensus. – Nate Eldredge May 07 '17 at 17:40
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    @NateEldredge so Chris would still be allowed to ask for feedback about the correctness of specific steps in his proof, just not general feedback about the correctness of the entire paper - am I understanding this policy correctly? – Dan Romik May 07 '17 at 17:43
  • @DanRomik: Yes, if such questions are at a research level. If the material being asked about is actually elementary (and this may be difficult for OP to judge, if he/she is not an expert), the question is not on topic and will be closed. – Nate Eldredge May 07 '17 at 17:44