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Publishing a Conjecture or Proof, from Due Diligence to Final Publication

Date: 06/16/2011 at 16:02:25
From: Randy
Subject: proofs and conjectures

1. Is there any listing(s) of proven, not yet proven, or disproved math
conjectures? If so, where would I find it?

2. What are the procedures for a person to submit a conjecture, or a proof 
of an as-yet unproven conjecture?



Date: 06/16/2011 at 19:35:02
From: Doctor Vogler
Subject: Re: proofs and conjectures

Hi Randy,

Thanks for writing to Dr. Math. Good question!

If you are under the impression that there is some kind of International
Bureau of Mathematics -- a global organization that registers every
conjecture, checks proofs, and assigns credit to the first mathematician
who submits a correct one -- allow me to disabuse you now. There is, in
fact, no such central authority.

Most (though not all!) original mathematics is done by professors of
mathematics at various colleges and universities around the world.
Regardless an author's affiliations, new proofs of previously-unproven
theorems are typically submitted to a peer-reviewed or refereed journal,
which is a scholarly publication that is printed on a periodic basis.

There are many journals of mathematics: some printed in one language while
others may contain articles in several languages ... some more prestigious
or well-read than others ... some specializing in a specific branch of
mathematics while others are more general ... some not printed at all
(though probably accessible online after paying a subscription fee) and
necessarily newer (and likely less prestigious) ...

When a mathematician discovers a proof for a well-known conjecture, she
writes up the proof in a paper (typically using LaTeX) and submits it to
an appropriate math journal. The journal employs reviewers who read the
article, check it for mistakes and such, possibly suggest changes to the
author, and (if the article is accepted) eventually print it in their
journal.

In this modern world, the article is often made available by the author as
a "pre-print" on her website or a repository such as arXiv before it is
published. While many pre-prints are largely ignored, pre-prints of
important results (such as claims of proofs of important well-known
unproven conjectures) are generally read by many mathematicians very
early, which usually results in incorrect proofs getting culled quickly.
After publication, the journal generally has some control of the
publication of the article, so you may have to get the final version
through the journal exclusively.

Math journals will rarely publish an article which is simply a conjecture.
Named conjectures generally appear in articles where the author presents a
new idea and develops much of the theory, or presents a new theorem along
with a complete proof, and then the author suggests some further work in
the area that may be worth pursuing, such as conjectures that appear to be
true but which the author has left unproven. The author will rarely put
her own name on the conjecture, and it only gets her name when subsequent
authors refer to her conjecture by her (which, alas, is how many theorems
and conjectures become falsely attributed to the wrong mathematician).

While an advanced textbook on some subject will generally mention many
important unproved conjectures related to that subject, complete lists
do not, unfortunately, exist. Having said that, you may be able to find
someone who has attempted to collect and list many such unproved
conjectures (which mathematicians usually call "open problems"), or open
problems in some particular field of mathematics. One reason for this is
that it's a moving target, since new results come out all the time.
Another reason is that there is no universal place to deposit unproven
conjectures. (If there were such a place, it would quickly be flooded with
unreasonable numbers of mostly-silly conjectures. After all, there is a
lot more that we don't know than there is that we do.)

But I think that the main reason is that much of mathematics is done at
the esoteric fringes, where mathematicians study ideas that may be
important to some small group of people, but which most people would not
care about in the slightest. Indeed, many (arguably most) published papers
require a solid understanding of some esoteric branch of math to grasp
what the conjecture even deals with, so listing the conjecture anywhere
except in a paper or book dedicated to that narrow subject is just silly.

In answer to your second question: a mathematician who believes that he
has solved an open problem will generally first have his proof or paper
checked by at least one other mathematician who specializes in the field
that the open problem belongs to, and then he will select an appropriate
math journal and submit his paper to the journal, which will review the
paper and then publish it in their journal.

On a related note, there is an interesting article about claims of
solutions of long-standing open problems here:

    http://www.scottaaronson.com/blog/?p=304 

If you have any questions about this, please write back and I will try
to offer further advice.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/ 
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