proof of all positive integers are polite numbers except powers of two
Proof. Let k be a positive integer.
Let for . Let’s suppose that is a polite number. We can write
with and .
Multiplying both sides by two,
If is even, then is odd and then would have an odd factor, contradiction. If is odd, then is odd and then would have an odd factor, contradiction. Then, is not a polite number.
If , then is a polite number trivially.
Now, suppose that is not a power of two. Then we can factor as with odd, . Then,
If then is clearly a polite number.
so is a polite number.
|Title||proof of all positive integers are polite numbers except powers of two|
|Date of creation||2013-03-22 18:42:47|
|Last modified on||2013-03-22 18:42:47|
|Last modified by||n847530 (22696)|