2023 Author: Bryan Walter | [email protected]. Last modified: 2023-05-21 22:24
The number 42 is known not only for being the answer to "The main question of life, the universe and all that", but also because it was the last natural number less than 100, which could not be decomposed into the sum of three cubes. Now this question has been resolved, since mathematicians Andrew Booker and Andrew Sutherland found the required expression: 42 = (-80538738812075974) 3 + 804357581458175153 + 126021232973356313, the solution was published on the page of one of the mathematicians (the page code contains links to his work, and from the faculty website mathematics, there is a direct link to the personal page).
Equations, in which only whole numbers can be unknown, are called Diophantine in mathematics - after the name of the ancient Greek thinker who dealt with them. Despite the seeming simplicity and comprehensibility even for a high school student, Diophantine equations can be extremely difficult to solve.
The most famous Diophantine equation is certainly Fermat's Last Theorem xn + yn = zn for integers n> 2. This claim was proved in 1994 by Andrew Wiles, more than 350 years after the original formulation. The proof uses methods that are by no means elementary mathematics and is over 100 pages long.
One of the open problems in the field of Diophantine equations is the expansion of natural numbers into the sum of three cubes of integers, that is, solving an equation of the form k = x3 + y3 + z3 for different k. It is known that for k, which is divided by 9 in the remainder of 4 or 5, there can be no such expansions, so they are excluded from consideration. The hypothesis is that all other k can be decomposed into such a sum.
An interesting feature of this problem is the alternation of very simple solutions and extremely complex ones. For example, for k = 29 there is an obvious solution x = 3, y = z = 1, while for k = 30, the solution is achieved only for gigantic values x = 3982933876681, y = -636600549515 and z = -3977505554546.
During the second half of the 20th century, expansions in the sum of three cubes were found for almost all the sought numbers less than 100. In particular, in the 60s, expansions were found for 87, 96, 91 and 80, then for 39, 75 and 84, then for 30, 52 and 74. By 2019, there were only two numbers left: 33 and 42. For 33, a solution was found by Andrew Booker of the University of Bristol this spring.
Now, together with Andrew Sutherland of the Massachusetts Institute of Technology, Booker has found a solution for the number 42. Its expansion is as follows: 42 = (-80538738812075974) 3 + 804357581458175153 + 126021232973356313. It can be difficult to verify the correctness of this expression using ordinary calculators, but you can use the calculations online with WolframAlpha.
Now the smallest number that has not been decomposed into three cubes is 114. Among the numbers less than a thousand such numbers, by the way, are also not very many: 165, 390, 579, 627, 633, 732, 795, 906, 921 and 975.
The proof of Fermat's Last Theorem is very instructive - we talked about some interesting facts related to it in the material "Who does not shake the fields."