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Legends hold that once upon a time, some mathematicians were rather pleased about a 30-ish page result in functional analysis. Paul Erdos, upon learning of the problem, spent ten or so minutes thinking about the original problem, and came up with a two-line proof.

I believe I read about this first in the biography The Man Who Loved Only Numbers, and it seems as though the internet maintains this legend (q.v. here, here, here).

This seems extraordinary, which leads me to some skepticism. I cannot seem to find any other reference to the actual proof or the problem. Is this simply an urban legend? Did it actually happen? What was the problem, and what was the original 30-page result?

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Perhaps the margin wasn't big enough for a longer proof? – copper.hat Dec 18 '12 at 20:41
The story seems to be attributed to George B. Purdy, maybe it is worth sending him a letter? – Artem Dec 18 '12 at 20:43
@Artem Where did you find attribution? – Arkamis Dec 18 '12 at 20:45
I know Professor Purdy, I'll send him an email about it. – JSchlather Dec 18 '12 at 21:44
@gnometorule: The "open problems homework" anecdote actually happens to be true, if not as romanticized: – josh Feb 15 '13 at 4:17
up vote 55 down vote accepted

I sent Professor Purdy an email. I asked him what he recalled about the incident. With his permission I've copied his correspondence below.

Dear Jacob,

Yes, I was there, and I'm the one who told the story to Paul Hoffman, who then included it in his book "The man who loved only numbers."

The 30 page proof was written by Jack Bryan just before Erdos came to visit, at Texas A & M University. The problem was written on the blackboard in the mathematics lounge and Erdos saw it and asked "Is that a problem?" I told him yes, and he went over and wrote a two line proof on the blackboard. It's the most incredible thing I ever witnessed, and that's why I told Paul Hoffman the story. Ron Graham told Hoffman to talk to me because I knew Erdos well.

Much later it became obvious that Erdos loved this story, and I asked him how he did it without knowing the subject. He said, smiling, "Oh, I was a good student at school!"

I have come to realize also that Erdos was one of those mathematicians who could tackle an unknown area as if he knew it. He also prized this ability in others. He once demonstrated to me that Fred Galvin had this ability. Fred was at the board and Paul asked him a question, and he answered it, and he asked Fred "Had you seen these things before", and Fred answered that he hadn't, and Erdos turned to me and said "See!"


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Thanks for your effort! That is an amazing story! – Dave Dec 19 '12 at 22:22
One note: there may be a typo here. I could not find a Jack Bryan affiliated with Texas A&M, but I did find a Jack Bryant, affiliated with Texas A&M and the math department. – Arkamis Dec 20 '12 at 16:25

Yes, you are right. It was Jack Bryant, not Jack Brian. He might have retired by now. By the way, even before Erdos came to town, it was generally agreed that there must be a proof that was shorter than 30 pages, but not a two liner! Professor Don Allen talked to Erdos the next day to see if he could help with the research problem that had generated this result, but he reported to us later that unfortunately he couldn't. Don would doubtless remember the statement and proof of the two liner.

Professor George Purdy University of Cincinnati

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So what was the problem and what was Erdos' proof? – marty cohen Dec 22 '12 at 3:33
Dear Marty, I have just written to Jack Bryant to ask him. I will post his reply. – George Purdy Dec 24 '12 at 23:12
Sure would be nice to know what the result was (and the contents of Erdős's proof). Any update on that? – user43208 Dec 11 '14 at 12:42

There is another well documented story in which Erdos went to a seminar where an open problem in the topology of infinite-dimensional spaces was stated. I think it was something about determining the dimension (for a particular definition of dimension) of the rational points in an infinite-dimensional Hilbert cube, and that the problem or the seminar was by Hurewicz. Erdos did not solve it in seconds, but he did so very quickly on the spot, while the seminar was going on. The solution was by his own combinatorial methods.

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