> 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.
> > Assume n = 4k + 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.