Automatic Solving of Cubic Diophantine Equations Inspired by Ramanujan
By Shalosh B. Ekhad and Doron Zeilberger
.pdf
.ps
.tex
Written: July 30, 2020.
Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger
and arxiv.org
Abstract: In Ramanujan's Lost Notebook, he stated infinitely many `almost counterexamples' to Fermat's Last Theorem
for n=3, by solving X^{3}+Y^{3}+Z^{3}=1.
As often the case with Ramanujan, he gave no indication how he discovered it.
Using ingenious exegesis of Ramanujan's mind by Michael Hirschhorn, combined with
a much earlier `reading of Ramanujan's mind' by Eri Jabotinsky, we generalize and automate this process, and develop
a symboliccomputational algorithm
for solving cubic diophantine equations of the form aX^{3}+aY^{3}+bZ^{3}=c,
for any integers a,b, and some constant c (that depends on a and b).
Maple packages
Sample Input and Output Files for RamanujanCubic.txt

If you want to see an article with 641 theorems regarding solutions of cubic diophantine equations
of the from A X^{3} + B Y^{3} + B Z^{3} =CONSTANT for
A ≤ B < 100
The input gives you
the output.

If you want to see a book with 26 chapters regarding solutions of cubic diophantine equations
of the from A X^{3} + B Y^{3} + B Z^{3} =CONSTANT for
A ≤ B < 10, with detailed explanations and motivation,
The input gives you
the output.
Sample Input and Output Files for Pell.txt
Sample Input and Output Files for RatDio.txt

If you want to see a computergenerated article with 100 random theorems concerning certain
cubic diophantine equations with 3 variables
The input gives you
the output

If you want to see a computergenerated article with 50 random theorems concerning certain
quartic diophantine equations with 4 variables
The input gives you
the output

If you want to see a computergenerated article with 20 random theorems concerning certain
quintic diophantine equations with 5 variables
The input gives you
the output
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