unambiguity of factorial base representation
For all positive integers , it is the case that
We first consider two degenerate cases. When or , the sum has no terms because the lower limit is bigger than the upper limit. By the convention that a sum with no terms equals zero, the equation reduces to or , which is correct.
Next, assume that . Note that
hence, we have a telescoping sum:
Adding 1, we conclude that
If and are distinct factoradic representations with and (i.e. not padded with leading zeros), then they denote distinct integers.
Let and let .
Suppose that and are distinct. Without loss of generality, we may assume that . Then, . By theorem 2, we have and . Because , we have . Because , we have . Hence, , so does not equal .
To complete the proof, we use the method of infinite descent. Assume that factoradic representations are not unique. Then there must exist a least integer such that has two distinct factoradic representations and with and . By the conclusion of the previous paragraph, we must have . By theorem 2, we must have . Let be the chosen such that but and let be the chosen such that but . Then and would be distinct factoradic representations of whose leading digits are not zero but this would contradict the assumption that is the smallest number with this property. ∎
|Title||unambiguity of factorial base representation|
|Date of creation||2013-03-22 16:38:35|
|Last modified on||2013-03-22 16:38:35|
|Last modified by||rspuzio (6075)|