Automatic Solving of Cubic Diophantine Equations Inspired by Ramanujan

By Shalosh B. Ekhad and Doron Zeilberger

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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 X3+Y3+Z3=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 symbolic-computational algorithm for solving cubic diophantine equations of the form aX3+aY3+bZ3=c, for any integers a,b, and some constant c (that depends on a and b).

## Maple packages

• RamanujanCubic.txt, a Maple package that emulates Ramanujan (and Mike Hirschhorn with help from Eri Jabotinsky) to automatically discover and prove infinitely many solutions to cubic Diophantine equations of the form

A X3 + B Y3 + B Z3 =CONSTANT

for arbitarary integers A and B. Ramanujan famoulsy did the case A=B=1.

• Pell.txt, a Maple package to solve Pell-like equations, in other words quadratic diophantine equations using the C-finite ansatz.

• RatDio.txt, a Maple package to generate solvable Diophantine equations using the C-finite ansatz. It also solves quadratic ones using brute force, but Pell.txt above is better.

## 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 X3 + B Y3 + B Z3 =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 X3 + B Y3 + B Z3 =CONSTANT for A ≤ B < 10, with detailed explanations and motivation,
The input gives you the output.

## Sample Input and Output Files for Pell.txt

• If you want to see the generating functions producing infinitey many solutions to Pell's Equation X2-nY2=1 for non-perfect-square n from 2 to 2499
The input gives you the output

• If you want to see the generating functions producing infinitey many solutions to the generalized Pell's Equation n1X2-n2Y2=CONSTANT for all n1 < n2 ≤ 100 for various small constants CONSANT, and where such things exist (i.e. exclusing pairs where is it trivially impossible)
The input gives you the output

• If you want to see the generating functions producing infinitey many solutions to the generalized Pell's Equation n1X2-n2Y2=CONSTANT for all n1 < n2 ≤ 300 for various small constants CONSANT, and where such things exist (i.e. exclusing pairs where is it trivially impossible)
The input gives you the output

Warning: this is a large file, of course the former file is contained in it.

## Sample Input and Output Files for RatDio.txt

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

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