From: israel@math.ubc.ca (Robert Israel) Subject: Re: f^n+g^n=h^n Date: 18 May 2000 23:25:18 GMT Newsgroups: sci.math Summary: [missing] In article <8g1235$aau$1@nnrp1.deja.com>, Serge Zlobin wrote: >How to prove that there are no such analytic functions f,g,h that >f^n+g^n=h^n (n>=3). In the case n=2 there are such functions: >cos^2(z)+sin^2(z)=1. I think it can be done with the help of Picard's >theorem. Below is a sci.math.research posting on the subject from 1996. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 -------------------------------------------------- Subject: Re: Entire solutions of f^2+g^2=1 Author: Harold P. Boas Organization: Dept of Mathematics, Texas A&M University Date: Wed, 03 Jul 1996 10:29:32 -0500 Robert Israel wrote: > > In article <4r9kv3$1ag7@hearst.cac.psu.edu>, Alan Horwitz writes: > |> I am interested in all entire solutions f and g to f^2+g^2=1. I remember > |> seeing this somewhere, but I cannot recall where. > I've also seen this before, in fact I recall assigning it as homework > to one of my classes, but I don't recall the source. The solutions are [ ... ] Robert B. Burckel gives some history about this problem in his comprehensive book An Introduction to Classical Complex Analysis, volume 1 (Academic Press, 1979). In Theorem 12.20, pages 433-435, he shows that the equation f^n+g^n=1 has no nonconstant entire solutions when the integer n exceeds 2; when n=2, the solution is as given by R. Israel in his post. In the notes on page 458, Burckel attributes the theorem to a 1916 paper of P. Montel, and he gives some further results and references. P. Vojta, in his post, mentioned work on this subject by Fred Gross. Some precise references to papers by Gross are: Further results on Fermat type equations and a characterization of the Chebyshev polynomials (with C. F. Osgood), J. Math. Anal. Appl. 121 (1987), no. 2, 317-324; Addendum in the same journal, 134 (1988), no. 1, 254-255. On the functional equation f^n+g^n=h^n and a new approach to a certain class of more general functional equations (with C. F. Osgood), Indian J. Math. 23 (1981), no. 1-3, 17-39. On the equation f^n+g^n=1. II. Bull. Amer. Math. Soc. 74 (1968), 647-648; Errata, same volume, 767. On the functional equation f^n+g^n=h^n, Amer. Math. Monthly 73 (1966), 1093-1096. -- ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Harold P. Boas mailto://boas@math.tamu.edu :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::