[Return to Main Page] [Mail a Message] [CECM Page]

The ISC is the Inverse Symbolic Calculator, a set of programs and specialized tables of mathematical constants dedicated to the identification of real numbers. It also serves as a way to produce identities with functions and real numbers.

The following tables are used in the ISC: elementary functions evaluated at rational points, hypergeometric functions with rational arguments, and simple numbers like 1/sqrt(7)+exp(1). There are more than 400 tables at the present moment holding a total of over 50 million constants. There is an INDEX to all the tables which explains each one of them. The first table is a set of 150 mathematical constants such as Pi, e, and sqrt(2) as well as all the known constants that have a name or a reference in mathematical literature, such as the Parking Constant and the Feigenbaum Constant. Each entry in the BASE table is identified with the name, function, references and, if possible, a simple Maple routine to calculate some digits.

The first level of analysis (known as the simple lookup) tries to identify the unknown constant against a list of millions of entries, like a reversed telephone directory. If that first attempt fails to identify it may be that the constant is in fact a simple transform of a known one. For that we use a smart lookup. For example, your constant could be the log of something known or it may just be the constant multiplied by a simple rational. The smart lookup performs over 200 different transformations. So for a given constant we transform it and then we compare the result to the list once again, this increases the number of possible matches to virtually billions of constants. As a test type your phone number (7 digits) and then try the simple lookup, if it fails to find something then try that same phone number by putting 1-area code-phone number in front of it (with a decimal point) and the 'smart' lookup. We bet you it will find something (!).

Generalized expansions is for generalized base developments, that is, generalized continued fractions, Egyptian fractions, Egyptian products, Sierpinski, Pierce and Engels developments, factorial base, base k, arithmetic bases and arithmetic functions. One of the components is gfun, a Maple package specialized in the identification of sequences, manipulation of power series and reconstruction of a generating function using a sequence of integers.

Integer relations algorithms uses primarily the LLL and PSLQ algorithms. In short this level can answer the question: Is my constant a linear combination of known constants , is X = a*Pi^2+b*sin(Pi/5)-c*sqrt(3), or is it an algebraic number of relatively small degree ? (We can actually tell you if the number is of degree 80 or less with coefficients of a prescribed size, we just need a good numerical precision.)
In this level we also use a model of the real numbers such as the Gamma model. In fact, many real numbers can be expressed with the Gamma function and derivatives with rational arguments. For example with GAMMA = G, then Pi = G(1/2)^2, Catalan constant = 1/16 * Psi'(1/4) - 1/16 * Psi'(3/4), where Psi is G'/G. There are numerous examples of numbers that can be represented with that function. By using a lattice algorithm and specialized tables of constants one can actually identify real numbers and even find identities. The model is of course not universal but it is large enough to serve our purposes. Other models are used as well.