麻豆传媒

Add three cubed numbers, and what do you get? It is a question that has puzzled mathematicians for centuries.

In 1825, a mathematician known as S. Ryley proved that any fraction could be represented as the sum of three cubes of fractions. In the 1950s, mathematician Louis Mordell asked whether the same could be done for integers, or whole numbers. In other words, are there integers k, x, y and z such that k = x³ + y³ + z³ for each possible value of k?

We still don鈥檛 know. 鈥淚t鈥檚 long been clear that there are maths problems that are easy to state, but fiendishly hard to solve,鈥� says Andrew Booker at the University of Bristol, UK 鈥� Fermat鈥檚 last theorem聽is a famous example.

Booker has now made another dent in the cube problem by , previously the lowest unsolved example. He used a computer algorithm to search for a solution:

33 = 8,866,128,975,287,528³ + (-8,778,405,442,862,239)³ + (-2,736,111,468,807,040)³

To cut down calculation time, the program eliminated certain combinations of numbers. 鈥淔or instance, if x, y and z are all positive and large, then there鈥檚 no way that x³ + y³ + z³ is going to be a small number,鈥� says Booker. Even so, it took 15 years of computer-processing time and three weeks of real time to come up with the result.

For some numbers, finding a solution to the equation k = x³ + y³ + z³ is simple, but others involve huge strings of digits. 鈥淚t鈥檚 really easy to find solutions for 29, and we know a solution for 30, but that wasn鈥檛 found until 1999, and the numbers were in the millions,鈥� says Booker.

Another example is for the number 3, which has two simple solutions: 1³ + 1³ + 1³ 聽and 4³ + 4³ + (-5) ³ . 鈥淏ut to this day, we still don鈥檛 know whether there are more,鈥� he says.

There are certain聽numbers that we know definitely can鈥檛 be the sum of three cubes, including 4, 5, 13, 14 and infinitely many more.

The solution to 74 was only , which leaves 42 as the only number less than 100 without a possible solution. There are still 12 unsolved numbers less than 1000.