The World’s Longest Mathematical Problem Takes 10 Billion Years To Read

US researchers have succeeded in solving a longer mathematical problem ever developed by a supercomputer . The answer to the Boolean problem of the Pythagorean triples , enunciated 35 years ago, is so long that a human being would take 10 billion years to read it.

With an incredible size of 200 terabytes – equivalent to all digitized texts held by the US Library of Congress – it is the largest mathematical proof ever produced, reports the CNRS, French national research center.

Marijn Heule, Oliver Kullmann and Victor Marek from the universities of Texas, Swansea and Kentucky, respectively, presented their work during the SAT 2016 international conference, which took place this weekend in Bordeaux. The calculations were obtained in the Stampede supercomputer of the University of Texas.

 The problem posits the starting point of the Pythagorean triples , that is, a set of three integers such that the sum of the square of the first with the square of the second number gives exactly the square of the third number.

The question is whether each whole number of red or blue can be colored so that no terna are all of the same color. For example, with items 3, 4 and 5, the numbers 3 and 5 should be blue and the 4, red. It has been shown that there are 102,300 ways to color whole numbers up to 7,824, but once reached the figure 7.825 it is impossible to have multicolored ternas. 

Facts vs theory

As The Nature writes:

The Pythagorean triples problem is one of many similar questions in Ramsey theory, an area of mathematics that is concerned with finding structures that must appear in sufficiently large sets. For example, the researchers think that if the problem had allowed three colours, rather than two, they would still hit a point where it would be impossible to avoid creating a Pythagorean triple where a, b and c were all the same colour; indeed, they conjecture that this is the case for any finite choice of colours. Any proof for more colours will probably be much larger even than the 200-terabyte 2-colour proof, unless researchers can simplify the case-by-case checking process with a breakthrough in understanding.

Although the computer solution has cracked the Boolean Pythagorean triples problem, it hasn’t provided an underlying reason why the colouring is impossible, or explored whether the number 7,825 is meaningful, says Kullmann. That echoes a common philosophical objection to the value of computer-assisted proofs: they may be correct, but are they really mathematics? If mathematicians’ work is understood to be a quest to increase human understanding of mathematics, rather than to accumulate an ever-larger collection of facts, a solution that rests on theory seems superior to a computer ticking off possibilities.

That did ultimately occur in the case of the 13-gigabyte proof from 2014, which solved a special case of a question called the Erdős discrepancy problem. A year later, mathematician Terence Tao of the University of California, Los Angeles, solved the general problem the old-fashioned way — a much more satisfying resolution.

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