19/09/2020
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The 65-year-old problem, x³+y³+z³=42, has finally been solved!
A team led by Professor Andrew Booker from the University of Brisol, who also groundbreakingly found a solution to three cubes whose sum is 33, and Professor Andrew Sutherlands from Massachusetts Institute of Technology (MIT) used the Charity Engine, a worldwide supercomputer program and tapped almost 400,000 home PC's to solve the integers that would satisfy the Diophantine Equation:
x³+y³+z³=42
The solution to this Diophantine equation is
x=-80538738812075974
y=80435758145817515
z=12602123297335631
Originally, the 65-year old problem came from the University of Cambridge where they looked for the solutions of a Diophantine Equation x³+y³+z³=k where k is a positive integer between 1 and 100 and x, y, and z are integers. Initially, some numbers have small solutions and mathematicians and programmers need not to make any programs to solve the problems. For instance, if k=3, then x=y=z=1. It has never been easy until the case for k=33 and k=42 became the most difficult and unsolvable.
Note that there are some integers between 0 to 100 that will not have solutions, or it cannot be written as a sum of three cubes. These are numbers that are in the form 9k+4 and 9k+5 where k is an integer (this can be proved using concepts from number theory). Thus, 4,5,13,14,22,23,31, and 32 cannot be written as a sum of 3 cubes.
There are so many questions in mathematics that are left unanswered up to this date. It is thanks to mathematicians today that we get to learn more about how specific objects interact and patterns involving such objects. Thanks to the knowledge contributed by mathematicians, we will be a step closer towards understanding more about life and the universe itself.
References:
Booker, A. (2019). Cracking the problem with 33. Research in Number Theory, [online] 5(3). Available at: [Accessed 8 September 2020].
Miller, S. (2019). Sutherland Helps Solve Decades-Old Sum-Of-Three-Cubes Puzzle. [online] MIT School of Science. Available at: [Accessed 8 September 2020].
Miller, S. (2019). The Answer To Life, The Universe, And Everything. [online] MIT News | Massachusetts Institute of Technology. Available at: [Accessed 8 September 2020].
Phys.org. (2019). Sum Of Three Cubes For 42 Finally Solved—Using Real Life Planetary Computer. [online] Available at: [Accessed 8 September 2020].
