Diophantus probably knew, and Lagrange[L] proved, that every positive integer can be written as a sum of four perfect squares. Jacobi[J] proved the stronger result that the number of ways in which aâ€¦ (More)

Abstract: By using the methods of Rowland and Zeilberger (2014), we develop a meta-algorithm that, given a polynomial (in one or more variables), and a prime p, produces a fast (logarithmic time)â€¦ (More)

Using a certification procedure for Abel-type sums, we present a computerized proof of Abelâ€™s identity. Abelâ€™s identity [1],[2] can be proved in many ways, including the elegant combinatorial methodsâ€¦ (More)

My guess is that, within fifty or hundred years (or it might be one hundred and fifty) computers will successfully compete with the human brain in doing mathematics, and that their mathematical styleâ€¦ (More)

John Horton Conwayâ€™s Cosmological Theorem about sequences like 1, 11, 21, 1211, 111221, 312211, . . . , for which no extant proof existed, is given a new proof, this time hopefully for good. One ofâ€¦ (More)

We prove an explicit formula for the number of n Ã— n upper triangular matrices, over GF (q), whose square is the zero matrix. Theorem. The number of n Ã— n upper-triangular matrices over GF (q) (theâ€¦ (More)

Remark. This identity is easily verified using the WZ method. It holds for all positive n, since it holds for n=1,2,3 (check!), and since the sequence defined by the sum satisfies a certain (homog.)â€¦ (More)