Automatic Solving of Cubic Diophantine Equations Inspired by Ramanujan

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 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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