Sum-of-Three-Cubes Solver

This repo contains a C program to list simple solutions (LSS) for: x³ + y³ + z³ = n

The sums of three cubes is a hard math problem, see https://en.wikipedia.org/wiki/Sums_of_three_cubes and https://en.wikipedia.org/wiki/Diophantine_equation for details. For a C# program please visit: https://github.com/mill1/SumOfThreeCubesSolver.

🔧 Requirements & Installation

Requires a C/C++ compiler only. Then execute in a terminal window:

> git clone https://github.com/fleschutz/LSS  # or download and unzip the ZIP file (click the green button)
> cd LSS
> cc -O3 -fopenmp main.c -o mode              # compiles with OpenMP API for multi-threading
> ./mode <NUMBER>                             # replace <NUMBER> by the mode number (see below)

Mode 1: Calculate a solution

This mode calculates the result of: x³ + y³ + z³ for the given values of x,y,z (e.g. executing: ./mode 1 1 2 3 returns: 1³ + 2³ + 3³ = 36).

NOTE: Big numbers are supported, e.g.: ./mode 1 -80538738812075974 80435758145817515 12602123297335631 returns: -80538738812075974³ + 80435758145817515³ + 12602123297335631³ = 42.

Mode 2: List no solutions

No solution at all exists for: n equal 4 or 5 modulo 9.

./mode 2 returns: no_solutions.txt. This mode uses function listNoSolutions() and took 0.116s.

Mode 3: List trivial solutions for positive numbers

This mode lists trivial solutions for positive numbers of x,y,z only (or: no subtraction, only addition).

./mode 3 returns: trivial_for_positive_numbers.txt. It uses the function listTrivialSolutionsForPositiveNumbers() and took 0.037s.

Mode 4: List trivial solutions for negative numbers

This mode lists trivial solutions for x,y,z having at least one negative number (or: at least one subtraction).

./mode 4 returns: trivial_for_negative_numbers.txt. It uses the function listTrivialSolutionsForNegativeNumbers() and took 4h.

Mode 5: List trivial solutions

This is mode 2 + 3 + 4 combined.

./mode 5 returns: trivial_solutions.txt and took 4h.

Mode 6: List nontrivial solutions

This mode lists nontrivial solutions by using a 'shotgun' algorithm in function listNontrivialSolutions(). The additional argument is an exponent defining a value range for x, e.g. the default value of 6 means: x=[10^6..10^7].

• ./mode 6 3 returns: solutions_for_10^3_range.txt (took 0.042s on a Core i9).
• ./mode 6 4 returns: solutions_for_10^4_range.txt (took 3.4s on a Core i9).
• ./mode 6 5 returns: solutions_for_10^5_range.txt (took 5min 23s on a Core i9).
• ./mode 6 6 returns: solutions_for_10^6_range.txt (took 2h on a Core i9).
• ./mode 6 7 returns: solutions_for_10^7_range.txt (took 13h on a Core i9).
• ./mode 6 8 returns: solutions_for_10^8_range.txt.

🏆 Nontrivial Solutions

These nontrivial solutions have been solved already:

• 30 = 3982933876681³ + (-636600549515)³ + (-3977505554546)³ (solved 1999 by Beck, Pine, Tarrant, and Yarbrough Jensen)
• 33 = 88661289752875283³ + (-87784054428622393)³ + (-27361114688070403)³ (solved 2019 by Andrew Booker)
• 42 = (-80538738812075974)³ + 80435758145817515³ + 12602123297335631³ (solved 2019 by Andrew Booker and Andrew Sutherland)
• 52 = 60702901317³ + 23961292454³ + (-61922712865)³
• 74 = (-284650292555885)³ + 66229832190556³ + 283450105697727³ (solved 2016 by Sander G. Huisman)
• 165 = (-385495523231271884)³ + 383344975542639445³ + 98422560467622814³
• 795 = (-14219049725358227)³ + 14197965759741573³ + 2337348783323923³ (solved by Andrew Booker)
• 906 = (-74924259395610397)³ + 72054089679353378³ + 35961979615356503³ (solved by Andrew Booker)
• NOTE: Use WolframAlpha or ./mode 1 with copy & paste to check the results.

The only remaining unsolved cases up to 1,000 are the following seven numbers:

• 114 = ?
• 390 = ?
• 627 = ?
• 633 = ?
• 732 = ?
• 921 = ?
• 975 = ?
• NOTE: Want to get famous? Just solve one of these numbers. May the force be with you 🖖

📧 Feedback

Send your email feedback to: markus.fleschutz [at] gmail.com

🤝 License & Copyright

This open source project is licensed under the CC0 license. All trademarks are the property of their respective owners.