email@example.com wrote in news:firstname.lastname@example.org:
> 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
Why it is impossible? You've shown that the _smallest_ counterexample can't be of the form 4k+1, but that doesn't apply to any other numbers.
You're confusing 'n' with its image under the operations.
> 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.
Same problem. n is 4k-1, not 8k-1 or 64k-1 or anything else. You're confusing using k as a parameter and using it to describe the form of the number.