Terence Tao, better at maths than a computer (Image: Steve Jennings/Getty Images for Breakthrough Prize)
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A MATHS problem previously tackled with the help of a computer, which produced a monster proof the size of Wikipedia, has now been made manageable by a human. Although it is unlikely to have practical applications, the result highlights the differences between two modern approaches to mathematics: crowdsourcing and computers.
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Terence Tao of the University of California, Los Angeles, has published a proof of the Erdős discrepancy problem, a puzzle about the properties of an infinite, random sequence of +1s and -1s.
In the 1930s, Hungarian mathematician Paul Erdős wondered whether such a sequence would always contain patterns and structure within the randomness. One way to measure this is by calculating a value known as the discrepancy. This involves adding up all the +1s and -1s within every possible sub-sequence. You might think the pluses and minuses would cancel out to make zero, but Erdős said that as your sub-sequences get longer, this sum would have to go up. In fact, he said the discrepancy would be infinite, meaning you would have to add forever.
Last year, Alexei Lisitsa and Boris Konev of the University of Liverpool, UK, used a computer to prove that the discrepancy will always be larger than two. The resulting proof was a 13 gigabyte file – about the size of the entire text of Wikipedia – that no human could hope to check.
Tao, with the aid of crowdsourcing, has used more traditional mathematics to prove that Erdős was right: the discrepancy is infinite no matter what sequence you choose ().
“The proof of the ErdÅ‘s discrepancy was reached using traditional maths with the aid of crowdsourcing”
Lisitsa praises Tao for doing what his algorithm couldn’t. “It is a typical example of high-class human mathematics,” he says. But mathematicians are increasingly turning to machines for help, a trend likely to continue.
This article appeared in print under the headline “Wikipedia-sized maths problem cut down to size”
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