In Memoriam Raphael Robinson

 From BICA (16) 1996:

Raphael M. Robinson

John Brillhart

In the fall of 1948, when I entered U.C. Berkeley as a freshman, the university was bursting with who were getting a free education under the G.I. bill. By great good luck in my second semester, I took a course in the theory of equations taught by Raphael. We used the newly published, outstanding book of Uspensky [8].  I was deeply interested in the course, for it contained such topics as complex numbers, solving cubic and quartic equations, Vieta's and Newton's formulas, symmetric polynomials, elimination theory with resultants and discriminants, and Sturm sequences.

Raphael wrote out his lectures in advance in a clear, strong hand with a fountain pen and then copied the notes onto the board during the lecture. The notes were clear, direct, and complete and I began to get a glimpse of a new level of understanding mathematics.

Raphael was a very sensible and deliberate man who required you to make sense, even if you weren't quite sure what you wanted to say. As a student I found this insistence rather intimidating, and since he was a large man as well, I was somewhat cowed for some years until I got to know him better and took steps to be minimally sensible. It was also helpful later to have his wife Julia there, because then I was at least talking to two people.

In 1951—52, my senior year, I took the two-semester number theory course from Raphael, in which we used no text, only his notes. This course changed my life. The course contained the usual basic material such as congruences and quadratic reciprocity...all fascinating. But there were also continued fractions and the unusual application to factoring integers, Liouville numbers, Lucas sequences and their application to primality testing, and a detailed examination of the new Erdös-Selberg proof of the prime number theorem.

The class was well aware that Raphael was involved with factoring and primality testing. One day, for example, he strode into the classroom and immediately wrote on the board:

NEWS FLASH! New Mersenne prime. 22,203 - 1. Found Tues. October 7, 1952 at 3 A.M. Gives 16th perfect number.

He was examining Mersenne and Fermat numbers using the SWAC, the Bureau of Standards Western Automatic Computer, at the Institute for Numerical Analysis at UCLA, where D.H. Lehmer was the director. Actually Raphael had never seen nor programmed a computer, but he astonished everyone by writing a SWAC primality testing program up in Berkeley which ran the first time. This was typical of the kind of careful and clever work that Raphael did. He had figured out how the machine worked from some notes, and after asking some questions, had punched the binary program cards and sent the deck of cards down to UCLA to be run.

The Lehmers, who were busy at the time, put the deck on a shelf, planning to debug the program when they had a free moment. Raphael (up in Berkeley) waited and waited but no word came. Finally, he phoned and found out that they hadn't even tried his program. His response was to send them a telegram: "RUN IT!" The graduate student at UCLA who assisted with the research and the computing was John Selfridge [2] [7].  Later Raphael programmed the IBM 701 at U.C. Berkeley to factor some Fermat numbers [3] and numbers of the form 2n ± 1 [4].  He also published results from some SWAC runs on primes of the form k • 2n + 1 [6]. (Also see [5]).

During the factoring part of the number theory course, each student was given his own number to factor. My number was N = 20354021 = 3253 • 6257. It was wonderful fun. In Wheeler Hall, where the Math Department was situated at that time, there was a spare room in which a crank-turned desk calculator was kept on which we could do the arithmetic and a box of Hollerith cards, the Elder-Lehmer sieve cards, for factoring.

We used Legendre's factoring method in which quadratic residues were generated by the simple continued fraction expansion of the square root of the number N being factored. These residues were factored and multiplied together and square factors dropped, giving smaller residues. You tried to do this in a way so that a very small residue was found, which was then used to pick out the corresponding card from the box of sieve cards. When the resulting stack of cards was held up to the light, light would shine through any hole that went all the way through, giving a possible prime factor of N (up to the limit of the cards).

In Raphael's courses he was famous for his problems. Students would sweat over them. They were solid problems whose solution often depended on some schrewd construction or artful trick that made you feel exhilarated if you got it and annoyed if you didn't, especially if you had worked hard on the problem. Raphael's solutions to the problems were instructive in both how to do problems and how to write them up. I'm reminded of the advice that George Polya gave about problems you give to a class: Don 't give students vapid soup to drink. Give them a good tough piece of meat to chew on!

Raphael was a man of sterling character. He had a gentle, but ironic sense of humor, nicely tinged with an appreciation of the absurd. He spoke in a tenor voice with a slight metalic edge on it. I once got him to laugh his hearty laugh at dinner at the Lehmer's by playfully jiggling a shaker next to my car, pretending to listen carefully to try tell whether it contained salt or pepper.

Another time in the summer of 1970 when Mike Morrison and I were at UCLA working on the continued fraction factoring method, we had gotten to the point where we thought the method would probably be powerful enough to factor the main outstanding number at the time, the 39-digit seventh Fermat number 2128 + l, which had been known to be composite for 65 years. I went up to Berkeley and mentioned to Raphael and Lehmer that I was beginning to think of turning the digits of the factors we were expecting to find into some kind of puzzle so that people would have to work to get them. Both Raphael and Lehmer looked quite skeptical we would get the factorizations... they'd believe it when they saw it. Raphael then burst out laughing and said, "You mean you're going to multiply the factors back together again?!" (The factoring method did work [1], but we were too eager to distribute the factors to encode them in any way.)

Throughout the years Raphael and his wife Julia were wonderful to know. Their pleasant humor, their decency, their acuity, and their concerns for truth and beauty were always an example for me. They were informal people, unassuming and down-to-earth, who were always genuinely interested in what other people were doing. Raphael gave to me, as did D.H. and Emma Lehmer, the hands-on tradition of computing in mathematics that gives insight through example and allows one to build a proper foundation beneath the equally important theoretical side of mathematics. He had such an elegant combinatorial touch, combined with a sense of minimal construction!

Raphael passed away in San Francisco on January 27, 1995 at the age of 83. The last time I saw him was at a dinner at his house. The guests were Emma Lehmer, Hugh Williams, and myself. Although Dick Lehmer and Julia were no longer with us and I felt their absence deeply, the conversation was relaxed and amusing. We discussed the problem of the deer coming into Raphael's back yard from Tilden Park next to his house and also the peculiarities of the English language. We all laughed at the old example:

Time flies like an arrow.
Fruit flies like an apple.

These days when I go back to Berkeley and walk in the hills full of trees and unique houses, I can't help but feel how fortunate I was to have grown up in this beautiful area and to have found at the university a man like Raphael whose teaching and learning were so suitable to my tastes and interests. Much that I have done since that time rests on what Raphael taught me. I will always be profoundly grateful to him for his efforts on my behalf and for his sharpness of insight and his warm humor.


[1] M.A. Morrison and J. Brillhart, A method of factoring and the factorization of F7, Math. Comp., 29 (1975), 183—205.

[2] R.M. Robinson, Mersenne and Fermat Numbers, Proc. AMS, 5 (1954), 842-846.

[3] R.M. Robinson, Factors of Fermat numbers, MTAC, 11 (1957), 21—22.

[4] R.M. Robinson, Some factorizations of numbers of the form 2n ± 1, MTAC, 11 (1957), 265-268.

[5] R.M. Robinson, The converse of Fermat's theorem, American Math Monthly, 64 (1957), 703-710.

[6] R.M. Robinson, A report on primes of the form k • 2n + 1 and on the factors of Fermat numbers, Proc. AMS, 9 (1958), 673—681.

[7] J.L. Selfridge, Factors of Fermat numbers, MTAC, 7 (1953), 274 275

[8] J. V. Uspensky, Theory of Equations, McGraw-Hill, New York, 1948.


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