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unambiguity of factorial base representation
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(Definition)
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Proof. 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

Proof. By definition,
Since each term of the sum is nonnegative, the sum is positive, hence
 . Using the fact that
 to bound each term in the sum, we have
By theorem 1, the right-hand side equals  , so we have
 , which is the same as saying
 . 
Theorem 3 If
and
are distinct factoradic representations with
and
(i.e. not padded with leading zeros), then they denote distinct integers.
Proof. 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. 
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"unambiguity of factorial base representation" is owned by rspuzio.
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(view preamble)
Cross-references: property, number, digits, conclusion, infinite descent, complete, without loss of generality, side, bound, representation, factoradic, telescoping sum, equation, upper limit, lower limit, terms, sum, integers, positive
This is version 11 of unambiguity of factorial base representation, born on 2007-01-30, modified 2007-01-31.
Object id is 8846, canonical name is UnambiguityOfFactorialBaseRepresentation.
Accessed 599 times total.
Classification:
| AMS MSC: | 11A63 (Number theory :: Elementary number theory :: Radix representation; digital problems) |
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Pending Errata and Addenda
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