Identities of the Function f(x,y) = x^2 + y^3

Abstract

Harvey Friedman asked in 1986 whether the function f(x,y) = x^2 + y^3 on the real plane R^2 satisfies any identities; examples of identities are commutativity and associativity. To solve this problem of Friedman, we must either find a nontrivial identity involving expressions formed by recursively applying f to a set of variables {x_1,x_2, ..., x_n} that holds in the real numbers or to prove that no such identities hold. In this paper, we will solve certain special cases of Friedman's problem and explore the connection between this problem and certain Diophantine equations.