More fun with mathematics. Yay!
In Algebra/Number theory, many numbers are considered "interesting" due to their properties.
For instance (only considering non-negative integers, and I'm only mentioning a few of their properties):
- 0 is the unit under addition: a+0=0
- 1 is the unit under multiplication: a*1=a
- 2 is the first prime, and also the only even prime number (which makes it a very "odd" prime).
- 3 is the first Mersenne prime. Mersenne prime numbers follows the formula 2^n-1, where n is a prime number, but not all numbers of that form is prime (most are not). Mersenne primes are interesting for several reasons. There is for instance a relation between Mersenne primes and perfect numbers (see below). If 2^n-1 is prime, then (2^n-1)*2^(n-1) is a perfect number. Mersenne primes are often used in pseudo-random number generators.
- 6 is the first perfect number. A perfect number is a number which equals the sum of its factors - excluding the number itself, but including 1. 6=1+2+3. The next perfect number is 28 (1+2+4+7+14).
- 4 is the smallest composite number, a number which can be factorized.
- 17 is the sum of the first 4 prime numbers (2,3,5,7).
- 30 is the smallest sphenic number (a product of 3 distinct primes)
- 42 is the 2nd sphenic number, and the 7th pronic number (numbers which are the product of two consecutive integers).
- 43252003274489856000 is the total number of possible configuration of the Rubic's cube
- 2^74207281 − 1 (a 22 million digit number) is (January 2016) the largest known prime number, and also the 49th Mersenne prime. BTW: While we know there is an infinite number of primes, we don't know if the number of Mersenne primes is infinite. There are good reasons to believe that there are infinitely many, but it hasn't been proven.
- (2^74207281 − 1)*2^74207280 is the 49th and largest known perfect numer (Jan 2016). It contains 44677235 digits.
So, are all integers interesting? In the "Penguin Dictionary of Numbers", 39 was listed as an uninteresting number in early editions of the book, but not now. So?
The question is of course meaningless, since "interesting" is a highly subjective concept. People who find math boring will probably find most numbers uninteresting. And I who really love math tend to find every number I come across interesting. Even the uninteresting ones. But we're here for the fun of it, so let's take a closer look and prove it.
Let's assume that the answer is no, which means that there is a set of integers which have no interesting properties. By the well ordered principle (which says that every non empty set of integers contain a smallest element), there is a smallest uninteresting number. But surely, being the first uninteresting number must be interesting. Which leads to a contradiction, and we may draw the conclusion that yes, all numbers are interesting.
While proof by contradiction is a valid approach, as I said, the question is meaningless. BUT I FIND IT FUNNY!!!!!
This, btw, is called the "Interesting Number Paradox", and of course there is a piece about it on Wikipedia:
https://en.wikipedia.org/wiki/Interesting_number_paradox.
pibbur who thinks he himself must be interesting, because if he wasn't, he would be the only uninteresting one, thus making him interesting.
PS. We could reformulate the concept of being an interesting number in a more objective way, by saying that an uninteresting number is a number not listed, for instance (According to Wikipedia) in an entry of the "On-Line Encyclopedia of Integer Sequences" (
https://oeis.org/). However, numbers classified by such definitions tend to change. PS.