A trio of mathematicians decided to make lemons into lemonade and made major headway on a problem that mathematicians have been thinking about for hundreds of years.
The three were just finishing a project and thinking about next steps when, late in March, two of them contracted Covid-19, separately but nearly simultaneously. Many people would take a break, but the third team member, Manjul Bhargava, thought differently. He suggested that they ramp up their weekly meetings to 3 or 4 times a week. The three were of the opinion that Quarantine could be an opportunity to think without fear.
One of the oldest questions in number theory was considered during these meetings. The number 6 can be written as (17-21) 3 + (37-21) 3.
Half of all the numbers can be written this way, according to mathematicians. The property seems to divide whole numbers into two equal groups, those that are the sum of two cubes and those that aren't.
Nobody was able to prove this or give any bound on the proportion of whole numbers in each camp. The camp consisting of sums of rational cubes may be vanishingly small or it may contain nearly every single number. If the Birch and Swinnerton-Dyer conjecture is true, about half of the numbers up to 10 million are the sum of two rational cubes, according to mathematicians. The rest of the number line can't be seen from such data.
The odd and even numbers are not the same as the two camps. No test is needed to determine which numbers belong in which camp. Both mathematicians can't prove that the test will always reach a conclusion, or they can't prove that the conclusion is correct
Number theorists have been embarrassed by the difficulty of understanding sums of cubes. His work on rational solutions to the elliptic curves of which sums of two cubes are a special case helped him win the Fields Medal.
In a paper posted online in October, Alpge, Bhargava and Shnidman showed that at least 2-21 and at most 1/6 of whole numbers can be written as the sum of two cubes.
There is a question of sums of cubes. Elliptic curves have a richly intricate structure that has propelled them to the center of many areas of pure and applied mathematics. One of the Clay Mathematics Institute's Millennium Prize Problems has a $1 million bounty on it's head.
Bhargava has developed a set of tools over the past two decades to explore the full family of elliptic curves. Peter Sarnak of the Institute for Advanced Study said that understanding sums of two cubes meant analyzing a smaller family.
The family seemed too far out of reach. I would have told them that it looks too hard.
The sum of two squared fractions seem to be plentiful, but not any of the numbers are the sum of two squares. By the early 1600s, Albert Girard and Pierre de Fermat had figured out a simple test for determining which whole numbers are the sum of two squares. The sum of two squared fractions is what your number is. 490 factors into 2 1 5 1 7 2 7 is the only factor that has a rest of 3 when you divide by four. The sum of two squares is called 490.
The majority of numbers don't pass the test. The chance that a whole number is the sum of two squared fractions is zero. The same is true for sums of two fractions raised to the fourth power, or the fifth power, or any power higher than three, according to mathematicians. There is an abundance only with the sum of cubes.
The equations are different from those of all other powers. One of the things that makes the sum-of-two-cubes equations easy to understand is the fact that they have no rational solutions or infinitely many. The equations with the highest coefficients have a limited amount of rational solutions.
Rounding equations have finitely many solutions or none at all. The phenomena that are never seen in other settings are displayed in these equations. Cubes are not the same in all respects.
cubes are not easy to comprehend. There is no overarching method for counting the rational solutions that have been proven to work.
Even with all the computing power that we have, I don't know how many rational solutions it has.
The sum of two cubes can be enormous, as shown by the number 2,803, which is the sum of two cubes with 40 digits. Many of the fractions would involve more digits than could fit on a piece of paper.
Number theorists try to link elliptic curves with more tractable objects because of their ungovernable nature. While Alpge and Shnidman were fighting Covid, they and Bhargava built on work the latter had done with Ho and figured out a way to build at least one special. There was a plan to count the 2 2 2 2 matrices.
Two classical subjects that have been studied for more than a century were used to draw on. Geometric numbers involve how to count lattice points in different shapes. Over the past 20 years, the topic of elliptic curves has been enjoying a renaissance thanks to the work of Bhargava and his team.
The circle method was created in the early 20th century by the legendary Indian mathematician and his long time partner G.H. Hardy. The circle method and geometry-of-numbers techniques are being combined for the first time. It's very cool.
The trio was able to show that no 2 2 2 2 matrix exists for at least 1/6 of the whole number. The sum-of- cubes equation has no rational solutions for those numbers. The sum of cubes of two fractions can't be more than 1/6 of the total number of numbers.
They found that at least 12 of the numbers have the same matrix. It is tempting to think that the numbers are the sum of two cubes. The converse is that every number with a matrix is the sum of two cubes.
Alpge, Bhargava and Shnidman needed something that took information about a cubic equation and used it to make rational solutions. There is a flourishing subfield of the theory of elliptic curves, so the trio turned to two of the subfield's experts. The number must be the sum of two rational cubes if it has a single associated matrix. The paper contains a three-page appendix which is described as marvelous in itself.
They had to impose a technical condition that reduced the 5/12 subset down to 2-21, or 9.5%, of the whole numbers. Bhargava thinks that Burungale and Skinner will reach the rest of the 5/12 before too long. The techniques are getting stronger.
It will take a long time to prove that half of the numbers are the sum of two cubes. There are two numbers in this set that are the sum of two cubes and one that isn't. He said that handling such numbers would require new ideas.
Researchers are happy to have finally solved the question, and are eager to investigate the techniques in the proof further. The result can be explained very easily, but the tools are very advanced in number theory.
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