Life, the Universe, and Sum of Cubes: Finally an Answer for 42
In 1954, mathematicians from the University of Cambridge proposed a challenge to prove whether or not there are integers x, y, and z that satisfy the equation $x^3+y^3+z^3=n$ for each positive integer n. This type of equation, where all the variables must take integer values, is known as a Diophantine equation. Although this conjecture is easy to understand, the mathematical community has yet to give a complete answer. But mathematicians are one step closer to an answer with the discovery of a solution for $x^3+y^3+z^3=42 $by Andrew Booker and Andrew Sutherland: $(-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3 = 42$. The values for x, y, and z appear to be obscure, random numbers taken from thin air, but they are the result of 1.3 million hours of computing on a global grid of computers.
| For one set of numbers, it is relatively simple to show that no solutions exist, and that is when the number n has a remainder of 4 or 5 when divided by 9. This is because all cubes either have a difference of 1, -1, or 0 with the closest multiple of 9, so the sum of three cubes will have a difference between 3 and -3 with the closest multiple of 9. The small cases of n = 1, 2, and 3 were quickly solved, and the solutions of these cases could be used to find solutions for any n that was 1, 2, or 3 times a perfect cube. Around the 1910’s, mathematicians were able to construct solutions for slightly larger numbers using the solutions to n = 1, 2, 3; however, after these families of solutions were found, they were stumped –– humans simply didn’t have the computing power to solve larger cases. No major breakthroughs were made until 2009, when Andreas-Stephan Elsenhans and Jörg Janhel created an algorithm, based on the theories of Noam Elkies, that could search for all solutions with n less than 1000 and max ( | x | , | y | , | z | ) (the largest value amongst the absolute values of x, y and z) less than 10^14. With this algorithm, solutions were discovered for all n<100 that did not have a remainder of 4 or 5 when divided by 9 - except for 33, 42, and 74, three values that continued to elude mathematicians. |
| The solution for 74 was discovered by Sander Huisman in 2016, who extended the algorithm used by Elsenhans and Janhel to max ( | x | , | y | , | z | ) less than 10^15, but 33 and 42 remained unsolved, and many mathematicians theorized that there were no solutions for either number. Then, in February of 2019, University of Bristol mathematician Andrew Booker stumbled upon a YouTube video posted by Numberphile, a channel sponsored by the Mathematical Science Research Institute of Berkeley, California, that discussed the uncracked problem of 33. Booker decided to try his hand at solving the problem, and created an algorithm that relied on the minimum of | x | , | y | , and | z | instead of the maximum. In just three weeks, he found an answer: 33 = 8,866,128,975,287,528^3 +(−8,778,405,442,862,239)^3 +(−2,736,111,468,807,040)^3. Now, the only unsolved number under 100 was 42. Booker enlisted the help of MIT mathematician Andrew Sutherland. He soon realized that solving 42 would be more difficult and would require more computing power than a single computer, so he turned to Charity Engine, a program that links home computers with other computers across the globe to use their spare computing power. After 1.3 million hours of computing, Booker and Sutherland finally found the solution for 42. The only unsolved cases under 1000 that remain are 114, 165, 390, 579, 627, 633, 732, 906, 921, and 975, for which Booker and Sutherland are working on expanding their algorithm. |
Up until now, the decidability of the sum of three cubes problem remains unknown: Mathematicians still cannot prove whether or not there is an algorithm that can, within a finite amount of time, return a solution for any value n. This problem relates to the tenth problem of Hilbert’s 23 problems, which were proposed by David Hilbert in 1900 as a list of the most challenging and influential unsolved problem at the time. His tenth problem asked if there is an algorithm that could decide whether any Diophantine equation had a solution. In other words, is every Diophantine equation decidable? Hilbert’s tenth problem has been solved, and the answer is no; such an algorithm does not exist. The solution was the result of 21 years of combined effort by Martin Davis, Yuri Matiyasevich, Hilary Putnam, and Julia Robinson. While no general algorithm exists for Hilbert’s tenth problem, the sum of cubes problem remains open.
Solvable or not, the progress made on the sum of cubes problem thus far is impressive and serves as a reminder of how powerful the mathematics community is when it works together. It was a video produced in Berkeley, California that inspired a UK mathematician, an American mathematician, and a global grid of computers to find a solution for n = 42. With increasing links between mathematicians all around the world through the internet and the exponentially rising computing power we are capable of harnessing, the boundaries of what we can accomplish are constantly being expanded. Booker and Sutherland serve as proof that, when we harness innovation, we can truly solve the unsolvable.
References
About. (n.d.). Retrieved from https://www.numberphile.com/about.
Booker, A. R. (2019). Cracking the Problem With 33. ArXiv. doi: arXiv:1903.04284
Elsenhans, A.-S., & Jahnel, J. (2008). New sums of three cubes. Mathematics of Computation, 78(266), 1227–1230. doi: 10.1090/s0025-5718-08-02168-6
Houston, R. (2019, September 30). 42 is the answer to the question “what is (-80538738812075974)³ 80435758145817515³ 12602123297335631³?”. Retrieved from https://aperiodical.com/2019/09/42-is-the-answer-to-the-question-what-is-80538738812075974³-80435758145817515³-12602123297335631³/.
Huisman, S. G. (2016). Newer Sums of Three Cubes. ArXiv. doi: arXiv:1604.07746
Robinson, J. (2008, December). Hilbert’s Tenth Problem. Retrieved from http://www-history.mcs.st-and.ac.uk/Extras/Robinson_Hilbert_10th.html.
Starr, M. (2019, September 9). Mathematicians Solve ‘42’ Problem With Planetary Supercomputer. Retrieved from https://www.sciencealert.com/the-sum-of-three-cubes-problem-has-been-solved-for-42.
Sum of three cubes for 42 finally solved – using real life planetary computer. (2019, September 6). Retrieved from https://www.sciencedaily.com/releases/2019/09/190906134011.htm.