On the solution of certain equations with exponent sum 0 over Z2

Let G be a finite group. A Corollary of a result of Gerstenhaber and Rothaus [3] states that all equations in one variable over G with exponent sum nonzero are solvable over G. Lyndon [6] studied equations in one variable with exponent sum zero over Z(m). He showed that there was a relatively simple equation with no solution over Z(2). In this paper we will initiate a study of equations with exponent sum zero over Z(2). In particular we will show whether or not certain equations, which are generalizations of the equation studied by Lyndon, have solutions over Z(2). We are grateful to the referee whose suggestion for the case when p < 0 < q both simplified and generalized our original result. This paper constitutes part of the theses of the first and third named authors.