  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.
