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In an article by George Johnson in the New York Times back in 1999, it says that an amateur mathematician from India once sent Ian Stewart a proof of the Ramanujan-Nagell theorem that the Diophantine equation $x^2 + 7 = 2^n$ is solvable if and only if $n = 3, 4, 5, 7, 15$. The proof "was badly typed on strange paper and cast in an idiosyncratic style that would have given any journal editor the impression that the writer was a crank." However, it was correct, and after getting some help cleaning it up, the man published the proof.

To me, this is an inspiring story, and I would like to know the name of this man and to see the paper. I asked Ian Stewart but he said that he remembers the incident but not the identity of the man in question. I would try asking George Johnson but I am not sure how to contact him. I searched MathSciNet but was not able to guess which paper it was.

Does anyone know more details?

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  • $\begingroup$ Johnson's article is based on the book "Mathematical cranks" by Underwood Dudley. en.wikipedia.org/wiki/Underwood_Dudley Does the book have the name you are looking for? Let me mention that Dudley was sued by one personality mentioned in the book. I remember reading Richard Posner's decision a few years back - it is beautifully worded... $\endgroup$
    – Igor Pak
    Commented Jun 5, 2010 at 7:05
  • $\begingroup$ Unfortunately, Mathematical Cranks does not contain the word "Nagell" (according to amazon's search inside the book), so I'm afraid the answer is not there. $\endgroup$
    – John Stillwell
    Commented Jun 5, 2010 at 7:24
  • $\begingroup$ I have no details about this story, but George Johnson has a homepage which includes his [contact information][2]. You might want to send him an e-mail. (This answer should have been a comment, but I'm unable to post comments just yet) [2]: sciwrite.org/glj/email.html $\endgroup$
    – Liran Rotem
    Commented Jun 5, 2010 at 9:07
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    $\begingroup$ Having emailed George Johnson, he says that he does not know anything beyond what is said in the article. $\endgroup$
    – bhwang
    Commented Jun 6, 2010 at 5:01
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    $\begingroup$ Well, I hope George Johnson was not bothered by the fact that I emailed him as well. Of course he said the same thing in his reply to me. I'm wondering, as a matter of general MO etiquette, whether the person who asked the question in the first place should be assumed to be the one to contact a suggested person. Otherwise the poor recipient may be barraged by a slew of email messages all asking the same thing. $\endgroup$
    – Timothy Chow
    Commented Jun 6, 2010 at 18:03

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I'm going to argue that the story is apocryphal; there was no such proof by an amateur. Here's why

The linked story is not about the amateur and the proof; it's about mathematical cranks. The amateur is not named; all the article says is

Several years ago Dr. Stewart heard from a man -- in India again -- who had found a new, simpler proof for an obscure, pointless theorem in number theory written by Ramanujan and a collaborator. According to the Ramanujan-Nagell theorem, the only numbers one can square and add 7 to, yielding an answer that is a power of 2, are 1, 3, 5, 11 and 181. For example, squaring 3 and adding 7 gives 16, which is the fourth power (the square of the square) of 2.

Dr. Stewart was surprised to realize that the proof was correct, but it was badly typed on strange paper and cast in an idiosyncratic style that would have given any journal editor the impression that the writer was a crank. Dr. Stewart advised the writer to find an Indian number theorist who could teach him how to present a proper paper. Several years later the result was published, and soon after came another publication from the same man. ''It is worth reading these things occasionally,'' Dr. Stewart said.

A search on MathSciNet for any article between 1998 and 2003 containing both the words Ramanujan and Nagell turns up only 19 hits. None are the article in question.

A Google Scholar search with the same parameters turns up only 4 pages of hits. None are plausibly an article of a proof by one or two authors, (at least one from India)

The 2002 survey article Relevance of Srinivasa Ramanujan at the dawn of the new millennium by KS Rao talks about the Ramanujan-Nagell equation does not mention a new proof - one might expect that Rao would have known about this.

So what did happen? Where did the story come from? One could imagine that Stewart had some conversation with Johnson about mathematical cranks, and misremembered a story, which Johnson did not do enough checking to correct. (side note: Johnson writes regularly about science but has only written a couple of math articles)

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    $\begingroup$ When I wrote to Ian Stewart in 2010, he replied, "I recall the incident, but I don't have any information about it any more. We moved building some years ago and a lot got thrown out. Sorry." This neither proves nor disproves your theory but I thought it was worth mentioning. An "intermediate" theory is that everything about the story is correct except that the "new" proof of Ramanujan-Nagell turned out to be not so new after all once it was examined by an expert, and so it wasn't published. $\endgroup$
    – Timothy Chow
    Commented Dec 13, 2017 at 18:45
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    $\begingroup$ @TimothyChow I like your idea better - it fits all the facts. $\endgroup$
    – Stopple
    Commented Dec 13, 2017 at 19:09

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