Take the number 9. It can be expressed as the sum of 0, 1 and 8, which are respectively the cubes of 0, 1 and 2. Or take 17, which is 1 + 8 + 8, or the sum of the cubes of 1, 2 and 2. How many other numbers from 1 to 100 can be expressed as the sum of the cubes of three integers (whole numbers, positive or negative)?
This is a puzzle with its roots in 1954-55, when it was described by University of Cambridge mathematicians. It is not as easy as it may look. While 9 and 17 provide solutions with positive cubes, some numbers require negatives. For example, 11 is 27 – 8 – 8, which can be expressed as (– 8) + (– 8) + 27, or the sum of the cubes of – 2, – 2, and 3. Other numbers can be much trickier, requiring large cubes that include negatives. Such as 51, which is the sum of the cubes of – 796, 602 and 659, or (– 504,358,336) + 218,167,208 + 286,191,179.
As it turns out, not every number has a solution. During their search for solutions, mathematicians have deduced a rule showing that certain numbers cannot be expressed as the sum of three cubes. For the numbers that do not come under this rule, they kept looking for solutions, and found them one by one.
Just two solutions were proving elusive — for 33 and 42. In March this year, a solution was finally found for 33. This month, the same mathematician teamed up with another to find a solution for 42, putting the problem finally to rest.
The point of it all, if any
Why should it matter whether we can or cannot express a certain number as the sum of three cubes? “Mostly it’s just a bit of fun,” said Andrew Booker of the University of Bristol, the mathematician who worked on the solutions for both 33 and 42. “More seriously,” Booker added in his email to The Indian Express, “as number theorists, our interest in this sort of problem borders on philosophical, along the lines of ‘Is it even possible to solve this problem?’”
There are many mathematical problems that are easy to state but hard to solve; it has also been discovered that there are problems that are actually impossible to solve.
In March, the journal Research in Number Theory published Booker’s solution for 33 as the sum of three cubes, which he had found using a computer algorithm. Now, Booker and another mathematician, Andrew Sutherland of the Massachusetts Institute of Technology, have used the same algorithm to solve for 42.
Tough search and discovery
Some numbers can be expressed as the sum of three cubes in more than one way. For example, 10 is 1 + 1 + 8 (the cubes of 1, 1 and 2) and also 64 – 27 – 27 (the cubes of 4, –3, – 3).
For any integer, there is a conjectural formula for the average density of the solutions, Booker said. “For 33 and 42 that density is particularly low,” he said.
Booker spent weeks on a supercomputer before he found an answer for 33. For 42, Booker and Sutherland used Charity Engine, a crowdsourced platform that harnesses unused computing power from over 500,000 home PCs. It needed over a million hours of pooled computing, which translated into much less in real time. “We had some teething problems with getting the code up and running on their network, but once we got going it took less than a week to find the solution,” Booker said.
The number 42 is the sum of the cubes of (i) 12,602,123,297,335,631; (ii) 80,435,758,145,817,515; and (iii) minus 80,538,738,812,075,974. And 33 is the sum of the cubes of (i) 8,866,128,975,287,528; (ii) minus 8,778,405,442,862,239; and (iii) minus 2,736,111,468,807,040.
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