Computer-Generated Proofs of Mathematical Theorems

algorithm A recipe or set of instructions which, when followed precisely, will produce a desired result. binomial coefficient The binomial coefficient ( n k ) is the coefficient of x in the expansion of (1+x). It counts the number of ways of choosing k objects from a set of n objects. computer algebra system A computer package that enables the computer to do symbolic computations such as algebraic simplification, formal differentiation, and indefinite integration. Diophantine equation An equation in several variables for which only integer solutions are accepted. hypergeometric series A finite or infinite summation, 1 + a1 + · · ·+ ak + · · ·, in which the first term is 1 and the ratio of successive summands, ak+1/ak, is a quotient of polynomials in k. hypergeometric term A function of, say, k that is the kth summand in a hypergeometric series. proof certificate A piece of information about a mathematical statement that makes it possible to prove the statement easily and quickly. proper hypergeometric term A function of two variables such as n and k which is of the form: a polynomial in n and k times xy for some fixed x times a product of quotients of factorials of the form (an+ bk + c)! where a, b, and c are fixed integers.