Ten Problems in Experimental Mathematics

This article was stimulated by the recent SIAM “100 Digit Challenge” of Nick Trefethen, beautifully described in [12]. Indeed, these ten numeric challenge problems are also listed in [14], where they are followed by the ten symbolic/numeric challenge problems that are discussed in this article. Our intent was to present ten problems that are characteristic of the sorts of problems that commonly arise in “experimental mathematics” [14, 15]. The challenge in each case is to obtain a high precision numeric evaluation of the quantity, and then, if possible, to obtain a symbolic answer, ideally one with proof. Our goal in this article is to provide solutions to these ten problems, and in the process present a concise account of how one combines symbolic and numeric computation, which may be termed “hybrid computation,” in the process of mathematical discovery. The passage from object, α, to answer, Ω, often relies on being able to compute the object to sufficiently high precision, for example to determine numerically whether α is algebraic or is a rational combination of known constants. While some of this is now automated in mathematical computing software such as Maple and Mathematica, in most cases intelligence is needed, for example to choose the search space or to decide the degree of polynomial to hunt for. In a similar sense, using symbolic computing tools such as those incorporated in Maple and Mathematica often requires significant human interaction to produce material results. Such matters are discussed in greater detail in [14, 15].