Hirotaka Ebisui
Motomachi 4-12-10
Iwakunishi 740-0012
Japan
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.
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