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Coffman, McCormick, and Swinney made further measurements on the system, this time controlling the flow rate much more precisely. Again they found members of the MSS sequence, but some of the patterns occurred for three values of 7. At that time they knew nothing of our recent work on the indented trapezoid and suspected that their strange results were due to some systematic error. How they came to learn of multiplicity was a matter of pure chance. Swinney was visiting North Texas State (where Mauldin teaches mathematics) to give a talk. His hosts, looking for some way to amuse him between lunch and the colloquium, brought him to Mauldin's office. To pass the time, Dan started to discuss our discovery. Swinney immediately realized that what he and his colleagues had seen was not, after all, an artifact. They went on to identify the analogue of the indentation parameter c as trace impurities in one of the reactants. It is certainly gratifying when some purely mathematical construct helps to explain physical reality.

Number Theory

Stan Ulam's name seems to have disappeared from these pages; it is time to bring it back, if only briefly. Stan was not a number theorist, but he knew many numbertheoretical facts, some of them quite recondite. As all who knew him will remember, it was Stan's particular pleasure to pose difficult, though simply stated, questions in many branches of mathematics. Number theory is a field particularly vulnerable to the "Ulam treatment," and Stan proposed more than his share of hard questions; not being a professional in the field, he was under no obligation to answer them.

Stan was very much interested in "sieve" methods-the sieve of Eratosthenes to generate the primes is the most famous-but from an experimental rather than an analytic viewpoint. He was always trying to invent new sieves that would generate sequences of numbers that were in some sense prime-like. His greatest success was the "lucky number" sieve (the name is derived from a story in Josephus's History of the Jewish War). In Eratosthenes's sieve one crosses out 1 from a list of the integers and then, keeping 2 (the first prime), crosses out all of its other multiples. The first survivor after 2 is 3, so next one crosses out all of its higher multiples, and so on. In the lucky number sieve one first crosses out every second number, that is, all the evens; in fact, one throws them out of the list, which is consequently collapsed. The first survivor after 1 is 3, so, again starting from the beginning, one throws out every third number, collapsing the list further. The next survivor is 7, so one then throws out every seventh number, and so on. The first ten lucky numbers are 1, 3, 7, 9, 13, 15, 21, 25, 31, and 33. All lucky numbers less than 3,750,000 were known by the early sixties. (Compared to the sieve for the primes, the lucky number sieve is rather slow.) Perhaps progress has been made, but I doubt that the range has been increased by a factor of 100 to match our current knowledge of the primes.

Although the lucky numbers are clearly not a multiplicative basis for the integers, they do have some prime-like properties. For example, their asymptotic distribution is, to first order, the same as that of the primes (Hawkins and Briggs). The luckies are, however, somewhat sparser than the primes, as, if I am not mistaken, Stan predicted. (Expressions for pn, the nth prime, and ,, the nth lucky number are

Pn = n ln n + n ln(ln n) + higher order terms

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Of course these expressions make sense only for large n. Nevertheless, if we (recklessly) disregard the higher order terms, they imply that n > Pn for n > 1619. In fact, however, we find that > Pn for 11 ≤ n ≤ 3,750,000.) The distribution of the lucky numbers is similar to that of the primes in another respect: there seem to be an infinite number of lucky "twins," that is, luckies whose difference is 2. The evidence for this is far from overwhelming because the lucky sieve is hard to implement on a computer.

What I learned from Stan's ventures into number theory was that amateurs can make useful contributions to the field. That moved me to launch an attack on my favorite classical problem, the Goldbach conjecture. This is the statement, made by Christian Goldbach in a letter to his friend Euler, that every even integer equal to or greater than 6 is the sum of two odd prime numbers in at least one way. It remains unproven to this day, although very few mathematicians have doubts about its truth. Curiously, the analogous problem for odd integers, namely, that from some point on, each is the sum of three odd primes, was proved by Vinogradov in 1937. His original proof is long and difficult; it may have been at least a decade before its correctness was generally admitted. As for the Goldbach conjecture, the best result to date is that all sufficiently large even integers can be expressed as the sum of a prime and an integer that has at most two prime factors. This result, due to J.-R. Chen, is considered to be the greatest triumph ever achieved by sieve methods. That the Goldbach “property" is true for lucky numbers was conjectured by Stan, and work by Myron Stein and me gives some support. Stan's conjecture should not be too surprising in view of a 1970 study by Everett and me, which shows that almost all sequences with overall prime-like distributions have both the twin property and the Goldbach property. (Here "almost all" is to be understood in a measure-theoretic sense.)

In the mid sixties Myron Stein and I decided to look at the Goldbach problem numerically. We started by examining the so-called Goldbach curve, that is, the plot versus the even numbers of the total number of ways of expressing each as the sum of two primes. The curve is rather bumpy, usually peaking locally at multiples of 6 (as explained in the introduction to Stein and Stein 1964). Clearly many more primes exist than are necessary to constitute an additive 2-basis for the even numbers. This motivated us to look for sparse subsets of the primes possessing that property ("S” bases). We found a good algorithm (the S algorithm) for producing such subsets; each is completely determined by the choice of a smallest prime po. Our S bases cover almost all the evens from 2po to 10,000,000, leaving uncovered only a few low evens at the start. The sparseness achieved is striking; with one exception the S bases consist of less than 1.6 percent of the primes less than 10,000,000. The exception is the basis beginning with 7, which contains roughly twice as many primes as any other (a fact still unexplained). Our conclusion from this work is that the Goldbach property does not critically involve the famous prime property of being a multiplicative basis for the integers.

In concluding I must mention that the above investigation of the Goldbach problem moved the number theorist Daniel Shanks to convey on those involved the title "Los Alamos School of Experimental Number Theory." As to this new institution, there is no doubt that Stan Ulam was the founder. ■

Paul R. Stein has been a staff member at the Laboratory since 1950, working on problems that range from mathematical biology to nonlinear transformations and experimental number theory.


Further Reading

M. T. Menzel, P. R. Stein, and S. M Ulam. 1959. Quadratic transformations: Part I. Los Alamos Scientific Laboratory report LA-2305.

P. R. Stein and S. M. Ulam. 1964. Non-linear transformation studies on electronic computers. Rozprawy Matematyczne 39. Also in Stanislaw Ulam: Sets, Numbers, and Universes, edited by W. A. Beyer, J. Mycielski, and G.-C. Rota. Cambridge, Massachusetts: The MIT Press, 1974.

M. L. Stein and P. R. Stein. 1964. Tables of the number of binary decompositions of all even numbers 0 < 2n < 200,000 into prime numbers and lucky numbers. Los Alamos Scientific Laboratory report LA-3106, volumes 1 and 2.

M. L. Stein and P. R. Stein. 1965. Experimental results on additive 2-bases. Mathematics of Computation 19:427-434.

C. J. Everett and P. R. Stein. 1969. On random sequences of integers. Los Alamos Scientific Laboratory report LA-4268.

C. J. Everett and P. R. Stein. 1970. On random sequences of integers. Bulletin of the American Mathematical Society 76: 349-350.

R. G. Schrandt and S. M. Ulam. 1967. On recursively defined geometrical objects and patterns of growth. Los Alamos Scientific Laboratory report LA-3762. Also in Essays on Cellular Automata, edited by Arthur W. Burks. Urbana: University of Illinois Press, 1970.

N. Metropolis, M. L. Stein, and P. R. Stein. 1973. On finite limit sets for transformations on the unit interval. Journal of Combinatorial Theory 15:25-44.

Mitchell J. Feigenbaum. 1980. Universal behavior in nonlinear systems. Los Alamos Science 1(1):4-27.

Reuben H. Simoyi, Alan Wolf, and Harry L. Swinney. 1982. One-dimensional dynamics in a multicomponent chemical reaction. Physical Review Letters 49: 245-248.

William A. Beyer, Peter H. Sellers, and Michael S. Waterman. 1985. Stanislaw M. Ulam's contributions to theoretical biology. Letters in Mathematical Physics 10: 231-242.

W. A. Beyer, R. D. Mauldin, and P. R. Stein. 1986. Shift-maximal sequences in function iteration: Existence, uniqueness, and multiplicity. Journal of Mathematical Analysis and Applications 113: 305-362.

K. Coffman, W. D. McCormick, and Harry L. Swinney. 1986. Multiplicity in a chemical reaction with one-dimensional dynamics. Physical Review Letters 56: 999-1002.

J. D. Louck and N. Metropolis. 1986. Symbolic Dynamics of Trapezoidal Maps. Dordrecht, Holland: D. Reidel Publishing Co.

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tanislaw M. Ulam was one of a group of distinguished Polish mathematicians who immigrated

to the United States around 1939.

Other members of the group who come to my mind were Natan Aronszajn, Stefan Bergman, Samuel Eilenberg, Mark Kac, Otto Nikodym, Alfred Tarski, and Antoni Zygmund. Ulam was born in 1909 in Lwów, which was then within the boundaries of Poland. The society in which he grew up and was educated was almost completely obliterated during World War II; the surviving population was dispersed within the present boundaries of Poland. Lwów is now a Ukrainian city, and only its buildings remain to remind one that for several centuries it was an outpost of Polish culture.

Lwów was the birthplace of functional analysis, and Ulam, although he cannot

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be called a functional analyst, had strong connections to the group of mathematicians (Banach, Mazur, Steinhaus, and Schauder) who created it. Stan had a superb memory, and he beautifully described those times in his book Adventures of a Mathematician, in an article on the Scottish Café in the 1981 edition of The Scottish Book, and in an obituary for Stefan Banach. He received an excellent classical education at the Sixth Gymnasium in Lwów and throughout his life could quote Greek and Roman poetry. Encouraged by his parents to study "something useful," he went to the Lwów Polytechnic School to become an engineer, but he quickly became interested by a lively group of mathematicians and began to spend more and more time with them.

One of the people who attracted him

by Jan Mycielski

to mathematics was the famous topologist K. Kuratowski. Ulam told me about the following "little invention" he made while attending Kuratowski's course in calculus. Over a non-negative, decreasing function on the positive portion of the real line, build stairs of equal depth (Fig. 1). The problem is to prove that the shaded area (the sum of the areas between the function and the steps) is finite. Ulam said: "Shift the shaded pieces horizontally to the left until they all find themselves within the first column." Since the area of the first column is finite, the sum of the shaded pieces is also finite. Kuratowski was very happy to hear this original thought from his student. Perhaps that was the pebble that turned Ulam into a mathematician.

At the age of twenty, Ulam published his first paper: "A Remark on the Gener


alized Bernstein's Theorem." That short paper solves a problem posed by Kuratowski. It belongs to the theory of jigsaw puzzles (also called the theory of equivalence by finite decomposition) and is one of the earliest applications of graphs in set theory. It appears in the 1974 volume Stanislaw Ulam: Sets, Numbers, and Universes, which contains more than half of Ulam's hundred or so then-published papers. We can learn a lot from that volume. I will try to describe some of what I have learned, but first let me record some memories from our numerous conversations over the years.

Ulam liked to consider amusing objects and processes. It didn't matter to him whether or not they were real or imaginary, but they had to be intrinsically interesting, not just tools. Consequently most of his work has a directness similar to the directness of an observation of nature. That distinguishes his work from the majority of mathematical papers, which elaborate existing theories. In fact, in his later life he became quite critical of such mathematical investigations, which he regarded as too abstruse or unimaginative. He would even remark that the study of specific subjects, such as advanced chapters of algebra, algebraic topology, or analysis, was motivated by the history of mathematics rather than by the interest or notoriety of their problems. I would reply that mathematics is also an art, motivated by its internal beauty, and that only per

sistent study may reveal that beauty. He would agree only that his opinion was not easy to interpret correctly. In the end I am sure that there is wisdom in what he said, if only because he discovered several facts that are fundamental to modern mathematical culture, and I can hardly imagine discoveries of that nature in the areas he was criticizing.

Measurable Cardinals

I will now try to tell you about one of Ulam's important discoveries. It pertains to the foundations of mathematics and to the theory of large cardinal numbers. To give it the proper perspective, let me recall that Euclid was the first to organize the mathematics of his time into an axiomatic theory. That means he started from certain basic principles called axioms that he accepted without proof, and from them he obtained by pure deduction all the mathematical knowledge of his time. The system of Euclid became the accepted definition of mathematics until the time of Newton and Leibnitz. After the discovery of calculus, it became apparent that the development of mathematics within the system of Euclid is very unwieldy, and the system had to be abandoned. For a few centuries mathematics was in a sense unruly. Axiomatic organization returned to it around the turn of this century with the discoveries of Frege, Cantor, and Zermelo. Frege de


Fig. 1. K. Kuratowski posed the following problem to his calculus class: Over a nonnegative, decreasing function on the positive part of the real line, construct a step function with steps of equal depth. Prove that the area of the shaded regions between the two functions is finite. Ulam's solution was to move each shaded region into the first column, the area of which is finite.

veloped logic, Cantor invented and developed set theory, and Zermelo gave axioms for Cantor's set theory. Soon it became clear that all modern mathematics can be smoothly developed within set theory. Gradually it also became apparent that there is a whole hierarchy of larger and larger set theories, and one of the best ways to classify them is to see how large are the infinite cardinal numbers that can be shown to exist in those theories. (By a famous definition of Cantor, two sets A and B, finite or infinite, have the same cardinal number if and only if there exists a one-to-one function mapping A onto B). One might think that very large cardinal numbers are rather exotic and abstract objects whose existence is not of great mathematical interest. But by a famous theorem Gödel proved in 1931 (the socalled second incompleteness theorem), it follows that the larger the cardinal number whose existence can be proven in a given set theory, the more theorems can be proved in that theory, even theorems pertaining to such elementary operations as the addition and multiplication of integers. This fact was not yet known at the time Ulam made his discovery. His motivation was different-he was attracted by the mystery of the very large cardinal numbers for its own sake. Let me try to explain his theorem.

The smallest infinite cardinal number is called No (aleph zero). It is the cardinality of the set of integers. Clearly No

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