Douglas Hofstadter, Rutgers Mathematics Colloquium, April 11, 2014

See part 1 at

Abstract: In 1963, while playing novel number-theoretical games with a computer, I dreamt up a curious recursive definition for a real-valued function, which I dubbed "INT(x)". This function's behavior turned out to be very unpredictable. I spent a good deal of time exploring it computationally, and also invented and explored some variations on the theme. To my frustration, though, I was able to prove only the most basic facts about INT(x), and many intriguing questions remained completely unanswered, no matter how hard I worked. Eventually, as often tends to happen when one has pushed oneself as far as one can go and has hit up against one's limits, INT(x) slowly faded into the background of my life.

Many years later, my physicist doctoral advisor Gregory Wannier proposed, as my potential Ph.D. research, a canonical and very natural but unsolved problem concerning the mysterious energy spectrum of crystals in magnetic fields. I went for Gregory's bait hook, line, and sinker, but soon I found, to my great frustration, that unlike Gregory, who had made some small progress analytically, I was completely unable to prove anything analytically about the equation (known as "Harper's equation"). After a period of stagnation, finding myself in a box canyon with no escape route except using a computer to do experimental mathematics, I started exploring Harper's equation computationally, and to my astonishment, I found that good old INT(x) came back into the picture front and center. This was a delightful surprise, and all at once, out of the blue, my long-ago number-theoretical explorations turned out to give me some deep insights into the physics problem. However, despite all the progress, a lot of mystery remained (and still remains). In this talk, I will mainly describe Gregory Wannier's and my collaborative discoveries.

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