Computer cracks Erdős puzzle – but no human brain can check the answer

A puzzle that has confounded mathematicians for almost a century is closer than ever to being solved, it has emerged. But there’s one slight problem.

The calculations which prove a part of what’s known as “Erdős discrepancy problem” have been worked out by a computer. And the sheer amount of data – more than the entire contents of Wikipedia – is so vast that it would be practically impossible to be checked by a human brain.

The “discrepancy problem” was posed in the 1930s by renowned Hungarian mathematician Paul Erdős. It revolves around the properties of infinite sequences of numbers containing nothing but +1s and -1s. Patterns in such sequences can be measured by creating finite sub-sequences.

Professor Enrico Scalas, of the University of Sussex, explained the premise: “You have a sequence of 1s and -1s (for instance, generated by tossing a coin) and a constant C. One is looking for a finite subsequence long enough so that the sum of the elements of the subsequence is larger than C.”

The difficulty lies in actually proving this is the case mathematically. That’s where computers and their ability to perform complex calculations come in. With this aid, computer scientists Dr Alexei Lisitsa and Dr Boris Konev of the University of Liverpool managed to demonstrate that an infinite sequence will always have a discrepancy (the sum of the numbers in a sub-sequence) larger than two.

They took a sequence 1,161 numbers long and the resulting data was a 13-gigabyte file. That’s bigger than the 10GB estimated size of the entire written contents of Wikipedia. It is the start of solving the puzzle – but while computers have helped, they have not yet taken over from humans.

In a statement, the researchers told The Independent: “On the one hand, it is true that our computer-generated solution is beyond the reach of humans to fully understand. On the other, all we can say for now is that at the moment there is no known ‘better’ human-comprehensible solution – but it does not mean that such a solution could not (or will not) be found in the future.”

Mathematicians were philosophical about being beaten by a machine. Matt Parker, Public Engagement in Maths Fellow at Queen Mary, University of London, said: “The computer did the heavy-lifting, but it was the insight and creativity of its human programmers which made it possible.”

Chris Budd, Professor of Mathematics at the Royal Institution of Great Britain, added: “Computers are doing for maths what the printing press did for writing, in that they are opening up unlimited possibilities.”