Rational functions that satisfy fn + gn + hn = 1

Abstract

Solutions of non-constant rational functions of the Fermat type functional equations ( ⁎ ) f n + g n + h n = 1 , where n ≥ 2 are integers, are considered in the complex plane. It is known that ( ⁎ ) cannot possess non-constant rational function solutions when n ≥ 8 , and that there exist non-constant rational function solutions when 2 ≤ n ≤ 5 . The cases when n = 7 and n = 6 have been open questions for forty years. We give several formulas and restrictions that f , g and h must satisfy when f , g , and h are non-constant rational function solutions of ( ⁎ ) . As an application, we obtain lower bounds for the degrees of non-constant rational function solutions of ( ⁎ ) , which yield partial results on the forty-year old open questions when n = 7 and n = 6 , see Corollary 4.1 , Corollary 4.4 . We also obtain results of a similar nature for polynomial solutions of ( ⁎ ) .