another proof that a number is polite iff it is positive and not a positive power of
By definition, an integer is polite if it a sum of consecutive non-negative integers, itself must be non-negative. Furthermore can not be since a sum of at least two consecutive non-negative integers must be positive. So we may assume that is positive.
There are two cases:
is a power of :
Suppose that is polite, say , where is non-negative and , then
This means that is a power of , or and are both powers of by the unique factorization of positive integers. Since , , so that if were a power of , must be odd, which implies that is odd too. Since is a power of , this forces . As and , there is only one solution: and , or , showing that is the only power of that is polite.
is not a power of :
|Title||another proof that a number is polite iff it is positive and not a positive power of|
|Date of creation||2013-03-22 18:10:05|
|Last modified on||2013-03-22 18:10:05|
|Last modified by||CWoo (3771)|