We have a season in which the equation

** x3 + y3 + z3 = k, **

being k a positive integer, not to generate news (and what is most curious is the amount of media that are echoing of the question, unthinkable a few years ago, not too many). Also, two people are working on it, are the ones that fill all these news, ** Andrew Booker ** , University of Bristol, United Kingdom, and ** Andrew Sutherland ** , of the Massachusetts Institute of Technology (the famous MIT).

they Began by finding a solution for k = 33, mid-September, achieved a solution for k = 42 (the only value that is resisted between the first one hundred natural numbers) and a few days ago have found a new solution for k = 3.

** 5699368212219623807203 + (-569936821113563493509)3 + (-472715493453327032)3 = 3 **

you Already know two solutions for that value:

13 + 13 + 13 = 3

43 + 43 + (– 5)3 = 3

even Though the media only mentioned this new finding, it is true that in reality have been two, since they have also found another unknown to date

** (-74924259395610397)3 + 720540896793533783 + 359619796153565033 = 906 **

And one question, what do you bring those new to particular cases? In addition, theoretically, wouldn't there be an infinite number of solutions for each value of k?

Although there is some reason in that argument, what is certain is that the number of solutions does not arise from an immediate mode, as you might think, because of the rapid growth of the third power. The known values were small, single-digit, but there is a solution that is more up to numbers of that magnitude. That is one of the complications. And find these values gives clues about how to locate more, because now it is possible to estimate the growth factor.

The reason for that is k = 3 the value in which focus search is not casual: for k = 1, and k = 2, we have already found expressions which give infinite values, for any value of a and b are integers:

on the other hand, when one inquires into the mode in which they have discovered those numbers (it escapes no one that should not be easy to find values of the order, two of twenty-one digits and one of eighteen), and discovers that half a thousand of volunteers have given to use their computers to do the calculations, anyone who does not know the world of the computational investigation will be asked, and what is the merit of these two gentlemen? Obviously, ** the algorithm that is used. **

When one looks for this type of solutions, not brute force (that is, by trying all the numbers one-to-one), but that, in addition to apply theorems and properties of known mathematical, experiment with others that may be plausible, and which might allow in the future to compose a formal proof is correct. In a nutshell, these advances, although they may seem small, insignificant and useless to many people, are necessary and may determine the time to discover other more relevant and applicable. Science moves forward on the basis of small steps, but sure. Here there is space or opportunity for discoveries by chance or become rich and famous overnight (unfortunately the idea that this society conveys to the youth of today; let me use the famous phrase attributed to ** James Dean ** , slightly modified: to live fast and what it leads to is to die young and do not always leave a pretty corpse).

As a curiosity, the time that you have used these volunteers in discovering these values, if we were to use a single processor running continuously, has been to four million hours, or more than 456 years (I hope I have laid well the account; I leave 456.621 years, considering 365 days in a year). Obviously, taking into account the planned obsolescence (which is not legal, but it is testable), we never reached the solution with a single machine. On the other hand, the computer company is stuck, literally, for being chosen to carry out this type of work (for some it will be; there seems to be a tangible benefit and immediate, always reaching to something, of course). In this case, the lucky has been the software company Charity Engine to execute an algorithm.

well-Known solutions for all values of k less than the hundred, the search has not concluded. The next challenge (one of them) is to try to show that solutions exist for all values of k lower than one thousand. There are only nine cases to solve: 114, 165, 390, 579, 627, 633, 732, 921 and 975. If they try, do not try with values for x, y, z (in absolute value, or values, both positive and negative), lower than 10^15. It is already verified that, up to that order of magnitude, there are no solutions.

** Alfonso Jesus Population Saez is a professor of the University of Valladolid and member of the Commission of Disclosure of the Royal Mathematical Society of Spain (RSME). **

** The ABCdario Math is a section that emerges from the collaboration with the Commission of Disclosure of the RSME. **

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*Date Of Update: 02 October 2019, 15:00*