Sums-of-Three-Cubes Solver

This repo contains a C program to list simple solutions (LSS) for the equation: 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.

🔧 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 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 for x,y,z

This mode calculates the result of: x³ + y³ + z³ for the given values of x, y, z. Note the following examples:

• ./mode 1 1 2 3 returns: 1³ + 2³ + 3³ = 36
• ./mode 1 -80538738812075974 80435758145817515 12602123297335631 returns: -80538738812075974³ + 80435758145817515³ + 12602123297335631³ = 42

Mode 2 - List no solutions

This mode calculates and lists all non-existing solutions. No solution exists for: n equal 4 or 5 modulo 9.

./mode 2 returns: no_solutions.txt. This mode uses the listNoSolutions() function and took 0.116s on a Core i9.

Mode 3 - List all solutions for positive x,y,z

This mode calculates and lists all solutions for x >= 0, y >= 0, and z >= 0.

./mode 3 returns: solutions_for_positive_xyz.txt (for readability the file has been numerically sorted by executing: sort -g < infile > outfile). It uses the listSolutionsForPositiveXYZ() function and took 0.037s on a Core i9.

Mode 4 - List trivial solutions for negative z

This mode calculates and lists trivial solutions for z < 0.

./mode 4 returns: solutions_for_negative_z.txt. It uses the listSolutionsForNegativeZ() function and took 10s on a Core i9.

Mode 5 - List trivial solutions for negative y and z

This mode calculates and lists trivial solutions for y < 0 and z < 0.

./mode 5 returns: solutions_for_negative_yz.txt. It uses the listSolutionsForNegativeYZ() function and took 4h on a Core i9.

Mode 6 - List trivial solutions

This mode combines mode 2 + 3 + 4 + 5.

./mode 6 returns: trivial_solutions.txt and took 4h on a Core i9.

Mode 7 - List nontrivial solutions

This mode calculates and lists nontrivial solutions for a given value range of x. The value range is defined by an exponent, e.g. exponent 6 means: x=[10^6..10^7].

It uses a 'shotgun' algorithm in the listNontrivialSolutions() function.

• ./mode 7 3 returns: solutions_for_x_greater_10^3.txt (took 0.042s on a Core i9).
• ./mode 7 4 returns: solutions_for_x_greater_10^4.txt (took 3.4s on a Core i9).
• ./mode 7 5 returns: solutions_for_x_greater_10^5.txt (took 5min 23s on a Core i9).
• ./mode 7 6 returns: solutions_for_x_greater_10^6.txt (took 2h on a Core i9).
• ./mode 7 7 returns: solutions_for_x_greater_10^7.txt (took 13h on a Core i9).
• ./mode 7 8 returns: solutions_for_x_greater_10^8.txt (not finished yet).
• ./mode 7 9 returns: solutions_for_x_greater_10^9.txt (not finished yet).
• ./mode 7 10 returns: solutions_for_x_greater_10^10.txt (not finished yet).
• ./mode 7 11 returns: solutions_for_x_greater_10^11.txt (not finished yet).
• ./mode 7 12 returns: solutions_for_x_greater_10^12.txt (not finished yet).

🏆 Nontrivial Solutions

The following nontrivial solutions have been solved already in the past:

• 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 (solved by ?)
• 74 = (-284650292555885)³ + 66229832190556³ + 283450105697727³ (solved 2016 by Sander G. Huisman)
• 165 = (-385495523231271884)³ + 383344975542639445³ + 98422560467622814³ (solved by ?)
• 795 = (-14219049725358227)³ + 14197965759741573³ + 2337348783323923³ (solved by Andrew Booker)
• 906 = (-74924259395610397)³ + 72054089679353378³ + 35961979615356503³ (solved by Andrew Booker)

Use WolframAlpha to verify the solutions or execute: ./mode 1 x y z (copy&paste recommended). The only remaining unsolved cases up to 1,000 are the following seven numbers:

• 114 = ?
• 390 = ?
• 627 = ?
• 633 = ?
• 732 = ?
• 921 = ?
• 975 = ?

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.