An old puzzle for the new year

[LaurV]

Originally Posted by LaurV

If n contains no odd factor, Robert gave the demonstration why it can't be written as so.

If n contains an odd factor, charybdis gave the simplest algorithm to write it so.

The rest are complications (and only partially solve the problem, you all show how to write n if it have odd factors, but nobody except Robert show that no power of two can be written so. Maybe there are some HIGH powers of two which could be accidentally written so, therefore the second half of the proof is important too).

(edit: clarification: Robert also solved the "positive part", but charybdis' proof is simpler and constructive, however, charybdis, (as the others too), doesn't solve the negative part, i.e. "powers of two can't be written as so", (s)he only shows that powers of two can't be written as so by his/her method, which is no warranty that somebody else won't come with a new method that allows a particular power of two to be written as so).

[uau]

Originally Posted by LaurV

(edit: clarification: Robert also solved the "positive part", but charybdis' proof is simpler and constructive, however, charybdis, (as the others too), doesn't solve the negative part, i.e. "powers of two can't be written as so", (s)he only shows that powers of two can't be written as so by his/her method, which is no warranty that somebody else won't come with a new method that allows a particular power of two to be written as so).

That is not true. The second paragraph of charybdis's post was the other direction.

[LaurV]

Ok, you are right! (I had to go back and re-read that, it is correct!).

(sorry charybdis )