## 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.

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