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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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