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Mathematicians open up a new front for an old number problem

As a high In the mid-1990s, as a schoolboy, Pace Nielsen came across a mathematical question that he still struggles with today. But he’s not feeling bad: the problem that captivated him, the odd perfect number guess, has been around for more than 2,000 years, making it one of the oldest unsolved problems in mathematics.

Original story courtesy of Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to improve the public’s understanding of science by addressing research developments and trends in mathematics, physics and life sciences.

Part of the longstanding appeal of this problem stems from the simplicity of the underlying concept: a number is perfect when it̵

7;s a positive integer. nwhose divisors are exactly double the number itself, 2n. The first and simplest example is 6, since its factors – 1, 2, 3 and 6 – add up to 12 or 2 times 6. Then comes 28, whose divisors are 1, 2, 4, 7, 14 and 28 add to 56. The next examples are 496 and 8,128.

Leonhard Euler formalized this definition in the 18th century with the introduction of his sigma (σ) function, which sums the divisors of a number. For perfect numbers we have σ (n) = 2n.

Leonhard Euler laid down many of the formal rules by which mathematicians think about perfect numbers and manipulate them.Illustration: Jacob Emanuel Handmann

But Pythagoras was aware as early as 500 BC. Aware of perfect numbers, and two centuries later Euclid developed a formula to create perfect numbers himself. He showed that when p and 2p – 1 are prime numbers (whose only divisors are 1 and themselves), then 2p−1 × (2p – 1) is a perfect number. For example when p is 2, the formula gives you 21 × (22 – 1) or 6, and if p if 3, you get 22 × (23 – 1) or 28 – the first two perfect numbers. Euler proved 2000 years later that this formula actually generates every even perfect number, although it is still unknown whether the set of even perfect numbers is finite or infinite.

Nielsen, now a professor at Brigham Young University (BYU), was caught up in a related question: Are there odd perfect numbers (OPNs)? The Greek mathematician Nicomachus stated around AD 100 that all perfect numbers must be even, but no one has ever proven this claim.

Like many of his 21st century colleagues, Nielsen believes that there are likely no OPNs. And, like his colleagues, he doesn’t believe there is any evidence in the immediate vicinity. But last June he came across a new approach to the problem that could lead to further progress. It is the closest thing to the OPNs discovered so far.

A tightening network

Nielsen first learned about perfect numbers during a high school math competition. He immersed himself in literature and in 1974 came across a paper by Carl Pomerance, a mathematician at Dartmouth College, who proved that every OPN must have at least seven different prime factors.

“When I saw that progress could be made on this problem, I naively hoped that maybe I could do something,” said Nielsen. “That motivated me to go to number theory in college and try to move things forward.” His first paper on OPNs, published in 2003, further narrowed these hypothetical numbers. Not only did he show the number of OPNs with k Various prime factors are finite, as Leonard Dickson found in 1913, but the size of the number must be less than 24thk.

These were neither the first nor the last restrictions placed on the hypothetical OPNs. For example, James Sylvester proved in 1888 that no OPN could be divisible by 105. In 1960 Karl K. Norton proved that if an OPN is not divisible by 3, 5 or 7, it must have at least 27 prime factors. Paul Jenkins, also at BYU, proved in 2003 that the largest prime factor of an OPN must exceed 10,000,000. Pascal Ochem and Michaël Rao recently determined that an OPN must be greater than 101500 (and later pushed that number to 102000). For his part, Nielsen showed in 2015 that an OPN must have at least 10 different prime factors.

Pace Nielsen, a mathematician at Brigham Young University, has long studied odd perfect numbers. His latest work suggests a new way to determine if they really exist.Photo: Alyssa Lyman / BYU

Even in the 19th century there were enough constraints to make Sylvester conclude that “the existence of [an odd perfect number]- It escapes, so to speak, the complex network of conditions that include it from all sides – would be a small miracle. “After more than a century of similar developments, the existence of OPNs seems even more dubious.

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