Mathematics: An see the journal [7] and the books [4],[5]. Experimental Science In the following we give first a brief description of some of the useful tools in the armament of ex- Herbert S. Wilf perimental mathematics, and then some successful examples of the method, if it is a method. The ex- University of Pennsylvania amples have been chosen subject to fairly severe Philadelphia, PA 19104-6395, USA restrictions. viz. The use of computer exploration was vital to 1 The mathematician’s telescope • the success of the project, and Albert Einstein once said “You can confirm a the- the outcome of the effort was the discovery of ory with experiment, but no path leads from exper- • a new theorem in pure mathematics. iment to theory.” But that was before computers. I must apologize for including several examples In mathematical research now, there’s a very clear from my own work, but since I am most familiar path of that kind. It begins with wondering what with those, it seemed inevitable. a particular situation looks like in detail; it contin- ues with some computer experiments to show the structure of that situation for a selection of small 2 Some of the tools in the toolbox values of the parameters of the problem; and then comes the human part: the mathematician gazes 2.1 The CAS at the computer output, attempting to see and to The mathematician who enjoys using computers codify some patterns. If this seems fruitful then will find an enormous number of programs and the final step requires the mathematician to prove packages available, beginning with the two ma- that the pattern she thinks she sees is in fact the jor Computer Algebra Systems (CAS), Maple and truth, rather than a shimmering mirage above the Mathematica. Either of these programs will pro- desert sands. vide so much assistance to a working mathemati- A computer is used by a pure mathematician cian that they must be regarded as essential pieces in much the same way that a telescope is used by of one’s professional armamentarium. They are a theoretical astronomer. It shows us “what’s out extremely user-friendly and capable. there.” Neither the computer nor the telescope can Typically one uses a CAS in interactive mode, provide a theoretical explanation for what it sees, meaning that you type in a one line command and but either of them extends the reach of the mind the program responds with its output, then you by providing multitudes of examples that might type in another line, etc. This modus operandi otherwise be hidden, and from which one has some will suffice for many purposes but for best results chance of perceiving, and then demonstrating, the one should learn the programming languages that existence of patterns, or universal laws. are embedded in these packages. With a little In this article I’d like to show you some examples knowledge of programming, one can ask the com- of this process at work. Naturally the focus will be puter to look at larger and larger cases until some- on examples in which some degree of success was thing nice happens, then take the result and use realized, rather than on the much more numerous another package to learn something else, and so cases where no pattern could be perceived, at least forth. Many are the times when I have written lit- by my eyes. Since my work is mainly in combi- tle programs in Mathematica or Maple and then natorics and discrete mathematics, the focus will gone away for the weekend leaving the computer also be on those areas of mathematics. It should running and searching for interesting phenomena. not be inferred that experimental methods are not used in other areas; only that I don’t know those 2.2 Neil Sloane’s database of integer applications well enough to write about them. sequences In this space we cannot even begin do justice to the richly varied, broad, and deep achievements Aside from a CAS, another indispensable of experimental mathematics. For further reading, tool for experimentally inclined mathemati- 1 2 cians, particularly for combinatorialists, is examples are Neil Sloane’s On-Line Encyclopedia of In- n (3n + 4)!(2n 3)! teger Sequences, which is on the web at n!, (7n + 3)!, tn, − , <http://www.research.att.com/ njas>. This 7 4nn!4 now (early 2004) contains nearly 100,000∼ integer sequences and has full search capabilities. A great all of which are hypergeometric sequences. deal of information is given for each sequence. If you input the first several members of the un- known sequence, Rate will look for a hypergeo- Suppose you are trying to count a certain collec- metric sequence that takes those values. It will tion of objects of size n . Perhaps these are sets of also look for a hyper-hypergeometric sequence (i.e., n elements that have some property that interests one in which the ratio of consecutive terms is hy- you, or some property of the integer n, like the pergeometric), and a hyper-hyper-hypergeometric number of its prime divisors, that interests you, sequence, etc. etc. Suppose further that you’ve found the answer For example the line for n = 1, 2, 3,..., 10, say, but you haven’t been able to find any simple formula for the general an- Rate[1, 1/4, 1/4, 9/16, 9/4, 225/16] swer. Here’s a concrete example. Suppose you’re elicits the (somewhat inscrutable) output working on such a problem, and the an- 41−i0( 1 + i0)!2 . swers that you get for n = 1, 2,..., 10 are { − } 1, 1, 1, 1, 2, 3, 6, 11, 23, 47. The next step should be to look online to see if the human race has encoun- Here i0 is Rate ’s running index, so we would nor- tered your sequence before. You might find noth- mally write that answer as, perhaps, ing at all, or you might find that the result that (n 1)!2 you’d been hoping for has long since been known, − , (n = 1, 2, 3, 4, 5, 6) 4n−1 or you might find that your sequence is mysteri- ously the same as another sequence that arose in which fits the input sequence perfectly. Rate is a quite a different context. In the latter case, an part of the Superseeker front end to the Integer example of which is described below in section 3, Sequences database, discussed in 2.2 above. something interesting will surely happen next. If you haven’t tried this before, do look up the little 2.4 Identification of numbers example sequence above, and see what it repre- sents. Suppose that, in the course of your work, you en- countered a number, let’s call it β which, as nearly as you could calculate it, was 1.218041583332573. 2.3 Krattenthaler’s package Rate It might be that β is related to other famous math- ematical constants, like π, e, √2, and so forth, or A very helpful Mathematica package for guessing perhaps not. But you’d like to know. the form of hypergeometric sequences has been The general problem that is posed here is the fol- written by Christian Krattenthaler and is available lowing. We are given k numbers, α1,...,αk (the from his web site. The name of the package is Rate basis), and a target number α. We want to find (rot’-eh), which is the German word for “guess.” integers m,m1,...,mk such that the linear combi- nation To say what a hypergeometric sequence is let’s first recall that a rational function of n is a quotient mα + m α + m α + + m α (1) of two polynomials in n, like (3n2 + 1)/(n3 + 4). A 1 1 2 2 · · · k k hypergeometric sequence t is one in which { n}n≥0 is an extremely close numerical approximation the ratio tn+1/tn is a rational function of the index to 0. For, suppose we had a computer pro- tn=(n n. For example, if 7) then tn+1/tn works out gram that could find such integers, how would to be (n + 1)/(n 6), which is a rational function we use it to identify the mystery constant β = of n, so this is a− hypergeometric sequence. Other 1.218041583332573? 3. THINKING RATIONALLY 3 We would take the αi’s to be a list of the log- augmented by log 1.218041583332573, into Maple’s arithms of various well known universal constants IntegerRelations[LinearDependency] package. and prime numbers, and we would take α = log β. The output is the integer vector [2, 6, 0, 3, 4], For example, we might use which tells us that β = π3√2/36, to the− number of decimal places carried. log π, 1, log 2, log 3 (2) { } 2.5 Solving PDE’s as our basis. If we then find integers m,m1,...,m4 such that I had an occasion recently to need the solution to a certain partial differential equation that arose m log β + m1 log π + m2 + m3 log 2 + m4 log 3 (3) in connection with a research problem that was posed by Graham, Knuth and Patashnik in [10]. is extremely close to 0, then we will have found It was a first order linear PDE, so in principle that our mystery number β is extremely close to the method of characteristics should give the solu- β = π−m1/me−m2/m2−m3/m3−m4/m. (4) tion. As those who have tried that method know, it can be fraught with technical difficulties relating At this point we will have a judgment to make.