Quick question about prime factorization
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?
Last edited by Christopher Robin Davies; 04-26-2012 at 03:14 PM.