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 Here i0isRate ’s running index, so we would nor- Neil Sloane’s On-Line Encyclopedia of In- mally write that answer as, perhaps, teger Sequences, which is on the web at (n − 1)!2 <http://www.research.att.com/∼njas>. This , (n =1, 2, 3, 4, 5, 6) now (early 2004) contains nearly 100,000 integer 4n−1 sequences and has full search capabilities. A great which fits the input sequence perfectly. Rate is a deal of information is given for each sequence. part of the Superseeker front end to the Integer If you’re counting something, say as a function Sequences database, discussed in 2.2 above. of n, and you’ve found the first 10 values of your sequence, the next step should be to look it up 2.4 Identification of numbers online to see if the human race has encountered your sequence before. You might find nothing at Suppose that, in the course of your work, you en- all, or you might find that the result that you’d countered a number, let’s call it β which, as nearly been hoping for has long since been known, or you as you could calculate it, was 1.218041583332573. might find that your sequence is mysteriously the It might be that β is related to√ other famous math- same as another sequence that arose in quite a dif- ematical constants, like π, e, 2, and so forth, or ferent context. In the latter case, an example of perhaps not. But you’d like to know. which is described below in section 3, something The general problem that is posed here is the fol- interesting will surely happen next. lowing. We are given k numbers, α1,...,αk (the basis), and a target number α. We want to find integers m, m1,...,mk such that the linear combi- 2.3 Krattenthaler’s package Rate nation A very helpful Mathematica package for guessing mα + m1α1 + m2α2 + ···+ mkαk (1) the form of hypergeometric sequences has been written by Christian Krattenthaler and is available is an extremely close numerical approximation from his web site. The name of the package is Rate to 0. For, suppose we had a computer pro- (rot’-eh), which is the German word for “guess.” gram that could find such integers, how would A hypergeometric sequence {tn}n≥0 is one in we use it to identify the mystery constant β = which the ratio tn+1/tn is a rational function of 1.218041583332573? the index n. Thus We would take the αi’s to be a list of the log- arithms of various well known universal constants n (3n + 4)!(2n − 3)! n!, (7n + 3)!, tn, and prime numbers, and we would take α = log β. 7 4nn!4 For example, we might use are all examples of hypergeometric sequences. If {log π, 1, log2, log 3} (2) you input the first several members of the un- known sequence, Rate will look for a hypergeo- as our basis. If we then find integers m, m1,...,m4 metric sequence that takes those values. It will such that also look for a hyper-hypergeometric sequence (i.e., one in which the ratio of consecutive terms is hy- m log β + m1 log π + m2 + m3 log 2 + m4 log 3 (3) pergeometric), and a hyper-hyper-hypergeometric sequence, etc. is extremely close to 0, then we will have found that our mystery number β is extremely close to For example the line β = π−m1 /me−m2 /m2−m3/m3−m4 /m. (4) Rate[1, 1/4, 1/4, 9/16, 9/4, 225/16] At this point we will have a judgment to make. elicits the (somewhat inscrutable) output If the integers mi seem rather large, then the pre- sumed evaluation (4) is suspect. Indeed for any 1−i0 2 {4 (−1+i0)! }. target α and basis {αi} we can always find huge 3. RATIONAL THOUGHT 3 integers {mi} such that the linear combination (1) I used the Maple command pdsolve to handle the is exactly 0, to the limits of machine precision. The equation real trick is to get the linear combination to be ex- ∂u(x, y) traordinarily close to 0, while using only “small” (1 − αx − α0y) = integers m, mi, and that is a matter of judgment. ∂x ∂u(x, y) If the judgment is that the relation found is real, y(β + β0y) +(γ +(β0 + γ0)y)u(x, y), rather than spurious, then there remains the little ∂y job of proving that the suspected evaluation of α is correct, but that task is beyond our scope here. with u(0,y)=1.pdsolve found that For a nice survey of this subject see [3]. − γ (1 − αx) α There are two major tools that can be used to u(x, y)= 0 1+ γ discover linear dependencies such as (1) among β0 − β β0 1+ y(1 − (1 − αx) α ) the members of a set of real numbers. They β are the algorithms PSLQ, of Ferguson and For- is the solution, and that enabled me to find explicit cade [8], and LLL, of Lenstra, Lenstra, and formulas for certain combinatorial quantities, with Lov´asz [11], which uses their lattice basis re- much less work and fewer errors than would oth- duction algorithm. For the working mathemati- erwise have been possible. cian, the good news is that these tools are available in CAS’s.