About Ramanujan's Equations

Hirotaka Ebisui
ahib.ebisui@nifty.ne.jp
Motomachi 4-12-10
Iwakunishi 740-0012
Japan

Abstract

Equation x^3+y^3=z^3 is famous as Fermat's problem that has no natural-number solution. x^3+y^3=z^3+1, however, has nutural-number solutions. This equation is a special case of Ramanujan's Equation x^3+y^3=z^3+w^3. We used the Mapleéuprogram to find out natural number solutions to x^3+y^3=z^3+w^3=T. Only 600 solutions, therefore, are shown below in numerical order. These 600 solutions included a few ones to Case w=1.

On the way, we have known a general partial solution to Ramanujan's Equation x^3+y^3=z^3+w^3. In addition, we have also known a general partial solution to x^3+y^3=z^3+1

From these general solutions, we obtain x=383662070451, y=46411475668533, z= 46411484401224, w=34878367854 , T=99971538772614746324923301814093358719288 and 1440000^3+72001^3=1440060^3+1= 2986357263552216001.

We have shown the two general partial solutions. At the same time, we have obtained numerical value cases, using the general partial solutions and numerical expression processing software, Maple V. And we will discuss about the difficulties involved in searching for all solutions to Ramanujan's Equation.

©1999. ATCM, Inc