Fermat's Last Theorem, the Four Color Conjecture, and Bill Clinton for April Fools' Day

Abstract

1. INTRODUCTION. We often fool our students. We fool them by stating and proving theorems that are in a polished and final form. As a result, students are often unaware of the evolution of ideas, strategies, statements, and even important mistakes that inevitably lead to the beautiful theorems and they encounter. There is much value in understanding early ideas and mistakes. In some cases, mistakes lead to the discovery of powerful and deep new mathematical truths. Thus we believe that it is occasionally appropriate to celebrate mathematical mistakes and we cannot imagine a better time for such a celebration than on April Fools' Day. Here we describe and outline three erroneous from the l9th century. We provide proofs of Fermat's Last Theorem, the Four Color Conjecture, and the fact that one of us is Bill Clinton. This paper is based upon a special undergraduate mathematics colloquium the authors gave on April 1, 1996. 2. FERMAT'S LAST THEOREM. The problem that was to become one of the most infamous open problems in mathematics had a most inauspicious beginning. Circa 1637, Pierre de Fermat (Figure la), while studying Bachet's Latin translation of Arithmetica by Diophantus, came upon a discussion of the Pythagorean theorem. This inspired Fermat to write the following, now famous, lines in the margin: