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.Bruckman, Paul S. A proof of the strong Goldbach conjecture.

Internat. J. Math. Ed. Sci. Tech.39(2008), no. 8, 1102–1109.

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peer review – the titles alone should have subjected them to extremely close scrutiny – and that they ought to have been retracted long ago.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.