I recently discovered the following two papers:

Bruckman, Paul S. A proof of the Collatz conjecture. Internat. J. Math. Ed. Sci. Tech. 39 (2008), no. 3, 403–407, DOI: 10.1080/00207390701691574.

Bruckman, Paul S. A proof of the strong Goldbach conjecture. Internat. J. Math. Ed. Sci. Tech. 39 (2008), no. 8, 1102–1109, DOI: 10.1080/00207390802136560.

Mathematicians probably do not need to read any further to see where I am going with this.

For non-mathematicians, the Collatz and Goldbach conjectures are two of the most famous unsolved problems in mathematics. In addition to being the focus of active research, they are also popular targets for non-experts, who have proposed countless numbers of erroneous proofs over the years.

The MathSciNet reviews for the two articles (Collatz, Goldbach), which are written after publication by independent reviewers, identify a critical error in each, which invalidates their results.

The journal in question is apparently reputable (otherwise I would not bother) and is published by a major commercial academic publisher, though its main focus is on topics in mathematics education rather than pure mathematics. It appears that the chief editor has been in charge since before 2008. I did not find any errata or editor's notes regarding these papers. (A corrigendum of the Collatz paper was published, but it fixes minor typos only, and the MathSciNet reviewer apparently took these corrections into account.)

It seems to me that the papers cannot have ever undergone proper peer review - the titles alone should have subjected them to extremely close scrutiny - and that they ought to have been retracted long ago. But in light of their age, I wonder if it is it still appropriate to raise the issue with the journal's editor, or if people will just see it as "water under the bridge".

I am also not quite sure how to explain the issue tactfully. I think with most professional mathematicians, I could just show them the citations with no further explanation, and they'd immediately understand why this is bad. Obviously that didn't happen in the first place, but I'm not really sure how much more background I can give the editor without potentially appearing condescending or insulting.

I saw the question I found a published paper that looks dodgy. What to do? But I think this case is more egregious than that one, in that the papers are not merely "dodgy" but in fact flatly wrong, and frankly, an embarrassment to the journal. The two answers there suggest "comment on PubPeer" and "do nothing", neither of which seem adequate.

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    @tomasz: I don't really agree with that. Finding errors in mathematical work can be very difficult. If the paper is written clearly in a standard way by an experienced mathematician, then the right person will probably be led "by smell" to the trouble spot, but it may still take some work to identify the error. If it is written in an eccentric, non-standard or obscure way, then what any sentence means can be up for grabs. Anyway, I think what Nate means is "extremely close scrutiny if they want to publish it". Sep 11, 2016 at 20:54
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    Having said that, I read the MathSciNet review for the Goldbach Conjecture, and the author uses an identity in which a sum of values of an analytic (on (2,infinity)) function f over all primes p > 2 is exactly equal to the integral of f/(log x) from 3 to infinity, provided both sides converge. That just doesn't sound good, and as the reviewer points out, the first function f one might choose gives a counterexample. Sep 11, 2016 at 21:04
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    This case seems kind of special. Considering that Jeff Lagarias cites the Bruckmann paper on Collatz in his 2006 review "The 3x+1 Problem: An Annotated Bibliography, II (2000-2009)" as "This paper asserts a proof of the Collatz conjecture. However the argument given has a gap which leaves the proof incomplete. The erratum points out this gap and withdraws the proof." it seems like no action should be taken…
    – Dirk
    Sep 12, 2016 at 14:54
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    @Wildcard The issue here is that whatever is communicated to the editor, the message will be "you were grossly negligent in allowing these papers to be published in your journal". It is hard to imagine anyone being happy about that message. Sep 13, 2016 at 6:56
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    It is interesting to see that these articles are indeed retracted now. May 24, 2020 at 7:44

4 Answers 4


It is a strange case. You say that the author of the two papers is deceased. Given that and the other facts you have presented -- in particular both papers were published almost ten years ago, their flaws would be immediately suspected by any mathematician and are documented in the MathSciNet reviews -- it seems to me that the principal culprit and the principal victim are both the International Journal of Mathematical Education in Science and Technology (IJMEST).

The journal in question is apparently reputable (otherwise I would not bother)

Is it reputable, though? I hadn't heard of it before. Taking a look now, it is hard for me to tell.

and is published by a major commercial academic publisher,

Come now: we all know that being published by a major commercial academic publisher is no certificate of quality. Elsevier was caught publishing a several journals that were essentially reprinted advertisements for medical and pharmaceutical companies. That's memorably egregious, but other big companies have skullduggeries of their own. Conversely, big companies which do some really shady things also put out some really good journals, Elsevier being a good example. Taylor and Francis is a less good example: of the 46 journals they publish in mathematics and statistics, I only myself recognize three as being good...but three is enough. So major publishing companies publish good journals and bad journals: I don't think one can deduce much from this.

Back to the journal. I looked into IJMEST, and I find it a bit strange. The journal's aims and scope center it on (a kind of) mathematics and science education:

Contributions will be welcomed from lecturers, teachers and users of mathematics at all levels on the contents of syllabuses and methods of presentation. Increasing use of technology is being made in the teaching, learning, assessment and presentation of mathematics today; original and interesting contributions in this rapidly developing area will be especially welcome. Mathematical models arising from real situations, the use of computers, new teaching aids and techniques also form an important feature. Discussion will be encouraged on methods of widening applications throughout science and technology. The need for communication between teacher and user will be emphasized and reports of relevant conferences and meetings will be included.

I wanted to remark that I have a colleague who is (to say the least) active in the mathematics education field, so I know that the above description is not really that of a modern research journal in mathematics / science education. It is more along the lines of creating a shared community for mathematical teachers (which is also a worthy goal, and one that my colleague is interested in).

But the above description does not seem to be a good match for the papers in which IJMEST has published in recent years. These papers seem to be almost entirely in the field of mathematics itself: most papers state theorems and give proofs. Sometimes they knowingly give proofs of old theorems and their angle is to give new proofs, often accompanied by the claim that they will be easier for the student to understand (although in my experience as a mathematician it is common for new proofs to be accompanied by such claims). But many, perhaps a majority, of the articles, seem to be entirely devoted to mathematics, usually of the kind that is broadly understandable: e.g. lots of Fibonacci numbers. So the published work of the journal looks to be a lot closer to what is published by the Mathematical Association of America (MAA)...but without nearly as much attention to the quality of the exposition as given by articles published in MAA journals.

So something has gone terribly wrong for this journal to publish proofs of major open problems like Collatz or Goldbach: such papers should be out of scope of the journal. As @Corvus points out, submitting a paper claiming a proof of something like Goldbach to a journal like this just doesn't make any sense: such a paper should be published not only in a math journal but in one of the very top math journals, first because a correct proof of Goldbach merits publication there (such a proof would rank among the great mathematical achievements of all time!) and second because a top journal can get the top experts in the field to the vet the paper, which is needed in order for the community to accept the result.

When the editorial board of a journal like IJMEST receives a paper claiming a proof of a major conjecture, they should (I think) do one of the following things:

(i) Bounce the paper back immediately as being out of scope for the journal, or

(ii) Engage in a preliminary refereeing job to see if the paper looks serious. If so, they should reach out to the editorial board at an appropriate journal -- i.e., a top mathematics journal -- and try to do a handover of some kind.

I've thought it over for a while now (slow internet connection...), and although in general I believe strongly that any interested party can contact editors and try to get published results corrected, in this case I just don't see why that would be a helpful thing to do. I realize that I now believe that publishing short, fallacious proofs of two major conjectures in the same year is enough to irreparably damage the reputability of a mathematics education journal in my eyes. Either they are so far outside of the mathematical community that they don't understand the significance of problems like Goldbach to the mathematical community and how they have to be handled or they know and don't care: in particular, they don't really care whether the mathematics they publish is correct. I don't see how to fix either of those problems.

  • It is somewhat odd. I looked at the journal on MathSciNet and it seems to have some of its papers indexed (not all, unless it publishes a very irregular number of papers per issue), but the page for the journal has no indexing information, which I do not think I have encountered before. Sep 13, 2016 at 6:58
  • Here's some context: jcturner.co.nz/paul-s-bruckman-and-number-theory/…
    – PatrickT
    May 6, 2021 at 11:46

I would not recommend contacting the author, as

  1. The people who put together the disproofs probably already have done so, and
  2. You're likely to end up wasting time in futile argument.

I know a researcher who claimed to have solved P=NP. This researcher has done some nice work in other places, but this particular piece of work is based on hubris and naivete. Like many who have a personal area of blindness (as often happens with such famous problems), this researcher is pretty much impossible to talk to rationally about their pet subject. Thus, people basically just ignore it (which is easy since it's only ever been informally published as a preprint). With something that has been published, you can contact the editors and see if they'll act, but if they don't I would recommend the policy of simply ignoring the article rather than making it a crusade.

I recognize that this suggestion may be controversial: it just feels wrong, given scientific ideals, to let an incorrect result stand. The edges of science always have been and always will be cluttered with junk work that is wrong but not worth anybody's time to get retracted or corrected.

I find that this is best understood by the fact that significant results generally imply more than just the result itself. If a result both correct and meaningful, then there should be a great deal of intellectual productivity that can expand out from either the result or the machinery used to obtain it. If not, then an incorrect result is much like all the other numerous dead-ends of scientific inquiry that have resulted in true but apparently useless results. That changes, of course, if an incorrect paper is likely to mislead experts or to cause public harm, in which case it's worth fighting, but that doesn't seem to be the case to me here.

If possible, it's nice to be able to lay a retraction marker down so that nobody ends up wasting their time on it. That's really all that getting the retraction would mean, however: just decreasing the likelihood of people stumbling across it and wasting their time. The ultimate fate of these papers will change little, whether retracted or not.

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    "it just feels wrong, given scientific ideals, to let an incorrect result stand" -- in effect, "someone's wrong in an 8-year-old article" is a stronger version of "someone's wrong on the internet". Either one can easily stop you getting to bed by a reasonable time. Sep 11, 2016 at 23:48
  • It seems that Paul Bruckman was an active mathematician with some non-retracted publications, but towards the end of his life he published "proofs" of Riemann, Goldbach, Collatz, you name it. He died in 2003.
    – PatrickT
    Sep 18, 2020 at 23:37

I've dealt with this sort of thing in the past. I would begin by contacting the authors, explaining the problem and asking them to issue a retraction (while avoiding the temptation to ask why, if they really believed they had proven the Goldbach conjecture, they published the proof in an education journal!!).

If that fails, I would move on to the editorial board of the journal in question with the same request.

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    I have reason to believe the author is deceased. Anyway, in general I hesitate to engage with people who think they have proved the Goldbach conjecture - conventional wisdom is that such interactions are rarely productive and sometimes lead to continuing and unwanted correspondence. I agree that contacting the editor is a good idea; my question is how to phrase it. Sep 11, 2016 at 18:17
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    That makes sense regarding engagement with Goldbach-provers. I suppose I might write the editor and explain that by virtue of its public visibility the Goldbach conjecture attracts a great deal of attention from aspiring mathematicians [that's the nicest phrase I can think of for "crackpot"; you may have something better]. I'd go on to explain that if it had been proven, it would be a massive breakthrough in mathematics and a cause for massive international excitement. But instead, what they've published is a flawed proof; moreover a detailed explication of the flaw has been published...
    – Corvus
    Sep 11, 2016 at 18:55
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    ...at MathSciNet. This is a potentially embarrassing state of affairs for the journal, all the more so because of the prominence of the problem in question. I'd urge the editor to either retract the paper or obtain further review in light of the MathSciNet posting and, based on the outcome of that review, consider retraction at that stage. Most likely the editor will do nothing; "Internat. J. Math. Ed. Sci. Tech. retracts paper that should obviously never have been accepted" is hardly desirable press. But you can try. Likely the editor will do nothing. "Journal of
    – Corvus
    Sep 11, 2016 at 18:57

Out of curiosity, I clicked through to the original articles, and noticed that both were retracted by the journal's editor-in-chief.

Both notices refer to negative reviews and (unsuccessful) corrections elsewhere; neither mentions what initially brought this to the editor's attention.

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