On Thursday, July 25, 2013 7:06:40 AM UTC-7, ra...@live.com wrote: > The conjecture states that: > > > > Given a positive integer n, > > > > If n is even then divide by 2. > > > > If n is odd then multiply by 3 and add 1 > > > > Conjecture: by repeating these operations you will eventually reach 1. > > > > > > > > Proof: > > > > > > > > Let n be the smallest positive integer that is a counterexample to the conjecture. > > > > If n is even then it can be divided by two to give a smaller number, leading to a contradiction. >
If n is even, then it cannot be the smallest possible counter-example. > > Assume n = 4k + 1. > By definition n (even) = 3k + 1
> Multiply it by 3, add 1, and divide by 2 twice. > > > > The result is 3k + 1, a number smaller than n, leading to a contradiction. Therefore n has the form > > > > n = 4k - 1. > > > > Multiply by 3, add 1, and divide by 2. > > > > The result is 6k - 1. If k is odd, then 6k - 1 is one more than a multiple of 4, which is impossible, therefore k is even, and n has the form > > > > n = 8k - 1 > > > > Multiply by 3, add 1, and divide by 2. > > > > The result is 12k -1, with k necessarily even. In this manner it can be proved that n must have the form 16k - 1, 32k -1, 64k -1, and so on, requiring n to be infinitely large, which is impossible.