Thursday, June 30, 2016

Mathematical Trivia



You may know that June 28 is Tau Day, a celebration of the number 2π, which some people believe to be the superior circle constant. I didn’t know that it is the most mathematically perfect day of the year.
June 28 is a perfect date because 6 and 28 are both perfect numbers, numbers whose divisors add up to themselves. The numbers 1, 2, and 3 are the three divisors of 6 (other than 6 itself) and 1+2+3=6. Likewise, 28=1+2+4+7+14.
Euclid's Elements contains the first known theorem about perfect numbers. It says that if 2n-1 is prime, then (2n-1)×2n-1 is a perfect number. So 6 is (22-1)×22-1, or 3×2; 28 is (23-1)×23-1, or 7×4. Prime numbers of the form 2n-1, the numbers that generate these perfect numbers, are called Mersenne primes after the 17th-century French priest and math enthusiast Marin Mersenne. Almost all of the largest known prime numbers are Mersenne primes because they’re easier to find than other primes.
Euclid knew that every Mersenne prime gave us an even perfect number, but it also goes the other way: all even perfect numbers must come from Mersenne primes. This theorem was finally proved by Euler, and it's called the Euclid-Euler theorem, possibly maximizing the chronological distance between a theorem's eponyms. The odd perfect numbers are more mysterious. We haven’t found any yet, and if we do find one, it will have to be greater than 101500, but you never know what might be lurking around the corner.
Another mystery is whether there are infinitely many Mersenne primes, and therefore infinitely many perfect numbers, or if the Mersenne primes dry up somewhere down the number line, leaving us with only finitely many points of perfection.
For the time being, we’ve only found 49 Mersenne primes, so we know exactly 49 perfect numbers. On June 28, find one of those numbers. Get to know it. Appreciate its exponents, make friends with its factors. 
You can start small, getting cozy with one of the perfect numbers Euclid knew: 6, 28, 496, or 8128. Or you can reach for the stars and adopt the latest, greatest perfect number (274,207,281-1)×274,207,280, with its 44,677,235 digits, as your friend for the day.
Hence there are 49 choices.

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