Take any whole number X. Obviously it is either prime or composite. If it is prime, then by definition it has no factors other than itself and 1. If it is composite then it is the product of a unique series of smaller primes. That is the fundamental theorem of arithmetic. It seems to me that at least those prime factors has to be less than or equal to the square root of Y, where Y is the smallest perfect square that is larger than X. Right? In other words if X was 323, then the next perfect square is 324, which is 18 squared, and so at least one of 323’s factors has to be smaller than 18. (And that is correct in this case; 323’s factors are 17 and 19.)
That seems intuitively obvious to me and I have a vague memory of seeing the proof way back in junior high. But for the life of me I cannot remember what it is. Anybody know?
Simple proof by contradiction: assume that all of X’s components are larger than the square root of X. Multiply any two of them, and the result is larger than X. This contradicts the assumption, and so must be false.
(This strategy, assume what you want to disprove, is key in the proof that the square root of two is irrational. Simply assume it is rational – x/y – and show how this leads to a contradiction.)
If d divides n, then d and n/d cannot both be larger than the square root of n, else their product would be > n.
To illustrate, I am currently reading a book of 443 pages and I got curious whether that was prime. Well, obviously not divisible by 2,3, or 5. 7 is out since 450 is not divisible by 7. 11 is out by a simple test. 13 is out because 430 (443 - 13) is not divisible by 13. 17 is out because 460 is not divisible by 17 and for 19, I added 57 and saw that 500 is not divisible by 19. The next prime square is too large (529) so in under a minute I saw that 443 is indeed prime.
