Primes, Twin Primes, & Infinity
Most of us can semi-remember (can’t we?) the definition of a prime number from somewhere in our school days. A prime number is a (positive, whole) number that is only divisible by itself and 1. (Technically, the more precise definition is ‘has exactly two divisors, but there’s little difference as long as we remember that 1 is NOT considered a prime.) There are lots of things that could be explored further here, but those are for another day. 🙂
Interestingly enough, we KNOW that the primes ‘go on forever’ ! Another way to say that is that they are infinite. Another interesting way to say it is to note that there will then be NO largest prime!! (There will always be another one bigger!). This fact was proved by Euclid (of geometry fame) roughly 200 years before Christ. It is often considered one of the most beautiful and elegant proofs in mathematics (it can be understood by high schoolers) and that it should be over 2000 years old only adds to its intrigue. 🙂
Closely related to primes – and also of interest to number geeks – are twin primes. Twin primes are pairs of primes that are only 2 apart, like 5 & 7, 11 & 13, 101 & 103, etc. [FYI, 2 & 3 are both primes, but are not considered twin primes, since 2 is a unique and special case of a prime. (Why?)]
And here is the fascinating part: While primes themselves have been known to be infinite since well before Christ, the question of whether or not twin primes ‘go on forever’ is still unknown and unsettled!! No one has succeeded in proving (or disproving!) this fact to this day!! This is one of the beauties of math that the one question has been settled for so long, and yet the other (similar) question still goes unsettled!
BONUS Challenge:
If anyone out there can prove or disprove the above question for twin primes, it will get you a 6 FREE copies of (any/all of) my book(s)!! – AND make you (and me? :-)) very famous!! Good luck!!