James' World 2

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Mona Lisa’s smile. Mary Lou Retton’s Olympic vault. Mariah Carey’s musical pitch. All are considered perfect. So are the numbers 6 and 28.

With feats of artistry and athleticism, perfection lies in the eye of the beholder. But for numbers, perfection is mathematically defined. “Perfect numbers” are equal to the sum of their “proper” divisors (positive integers that divide a number evenly, not counting itself). For example, 6 = 3 + 2 + 1, and 28 = 14 + 7 + 4 + 2 + 1. While these mathematical curiosities are about as likely to grace the walls of the Louvre as they are to perform a twisting layout back somersault, they do offer something irresistible: a perfect mystery.

Euclid laid out the basics of perfect numbers over 2,000 years ago, and he knew that the first four perfect numbers were 6, 28, 496, and 8,128. Since then, many more perfect numbers have been discovered. But, curiously, they’re all even. No one has been able to find an odd perfect number, and after thousands of years of unsuccessful searching, it might be tempting to conclude that odd perfect numbers don’t exist. But mathematicians haven’t been able to prove that either. How is it that we can know so much about even perfect numbers without being able to answer the simplest question about an odd one? And how are modern mathematicians trying to resolve this ancient question?

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BIG MOUTH for Quanta Magazine

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What if black holes, those all-consuming gravitational behemoths of the cosmos, aren’t actually black at all—or even holes, for that matter? Instead, a new theory suggests black holes may be dark stars with hearts of extremely dense, exotic matter. This could help explain one of the biggest mysteries of the universe: the origin and nature of dark matter.

Black holes are real-life examples of Albert Einstein’s general theory of relativity taken to the extreme. They’re places in the universe where enormously dense amounts of matter stretch the fabric of space and time to its limit, forming an infinitely deep gravitational well that not even light can escape—hence the “black hole” name.

Scientists believe at the center of each black hole, there lies an infinitely small and dense point called a singularity. Gravity is so strong at the singularity that it forms an event horizon surrounding it, where the pull of gravity surpasses even the speed of light.

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A family eating hotpot at a restaurant in Beijing. China’s population is projected to contract sharply this century.Credit…Gilles Sabrié for The New York Times

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Amazon Sidewalk shares your Internet with smart homes — and surveillance devices. (Washington Post illustration/iStock)

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