Classic Unsolved Problems in Mathematics

Here are three of several classic unsolved problems in mathematics.  The problem statements are listed first, with definitions and background, as appropriate, below.

Mystery #1:  Are the pairs of twin primes infinite?  (Do they go on forever or end at some point?)

Mystery #2:  Are the perfect numbers infinite?  (Do they go on forever . . .?)

Mystery #3:  Are there any odd perfect numbers?

BACKGROUND:

Mystery #1:  Primes are positive integers (other than 1) that are divisible only by themselves and 1.  Twin Primes are primes that are only two apart:  3 & 5, 17 & 19, 101 & 103, etc.  Interestingly, it has been known for a long time that the primes themselves are infinite.  Euclid proved this roughly 200 years before Christ. So, this is one of those cases where it seems like it should be easy to settle, but no one has been able to prove that they go on forever or prove that they eventually quit.

Mystery #2:  Perfect numbers are those whose factors (other than the number itself) add up to the original number. Examples:  The factors of 6 (other than 6 itself) are 1,2, & 3.  1 + 2 + 3 = 6, so 6 is perfect.  28 is also perfect, since 1 + 2 + 4 + 7 + 14 = 28.  The next two are 496 and 8128, and these four were known to the ancient Greeks. There are currently only 48 known perfect numbers – and several of them are HUGE, but no one has succeeded in proving they go on forever or that they eventually stop.

Side Note:  Perfect numbers have a surprising and fascinating connection to Mersenne Primes (mentioned in earlier Math Tidbits), but this will have to wait for another Math Tidbit sometime.

Mystery #3:  All 48 known perfect numbers (see above) are even.  In fact they all end in 6 or 8.  It can be proven that if a number is perfect and even, it will end in 6 or 8, but no one has yet found an odd perfect number.  Nor has anyone (yet) proven that there can’t be any.