Last Thursday I turned 28. As I have been telling anyone who’ll listen, this is the last time I will be perfect (unless I live to be 496).

In case you don’t know: a number is called perfect if it is the sum of all its proper divisors. So 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14 are perfect. There is one even perfect number for every Mersenne prime. Precisely, if p is a prime such that 2^{p} - 1 is also prime, then 2^{p-1}(2^{p} - 1) is perfect (and obviously even). Furthermore, every even perfect number is obtained in this way. This fact is known as the Euclid–Euler theorem. It is unknown if there are infinitely many Mersenne primes, and therefore whether there are also infinitely many even perfect numbers. No odd perfect numbers are known at all. The web of restrictions is so tight that the existence of even one would seem like a kind of a miracle. Nevertheless miracles have been known to happen in number theory, and we cannot currently rule odd perfect numbers out.

Euclid’s direction of the Euclid–Euler theorem—that every Mersenne prime gives a perfect number—is entirely straightforward (sorry Euclid!). Euler’s proof that these are all the even perfect numbers is a little trickier, but could still be explained to bright high-school students. It isn’t though. Instead we make the poor buggers endlessly compute exact integrals that would never appear in real-world calculations anyway. If we want them to enjoy maths for its own sake, we should at least show them something beautiful.

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© Tom Harris 2015–2017.

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