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I believe this is false, even for larger (but still finite) values of $6$:

I believe this is false, even for larger (but still finite) values of $k$ (not $6$):

In a recent question about Fibonacci numbers, it was claimed that

1. Is the proof above correct? (If not, and the original claim is correct, can you give me a representation of the number 5473?) Edit: Please see Michael Lugo's answer for a paper which finds the representation with the fewest nonzero digits in this "signed Fibonacci base". Please consider the following the actual question here:

2. Assuming the proof is correct, is the original claim true for other non-integer bases? What I mean is the following:

In a recent question about Fibonacci numbers, it was claimed that

1. Is the proof above correct? (If not, and the original claim is correct, can you give me a representation of the number 5473?) Edit: Please see Michael Lugo's answer for a paper which finds the representation with the fewest nonzero digits in this "signed Fibonacci base". Please consider the following the actual question here:

2. Assuming the proof is correct, is the original claim true for other non-integer bases? What I mean is the following:

The first three operations reduce the number of terms in the expansion and thus strictly simplify the expression (in terms of how many terms there are), but the last may need to be used several times before it "simplifies" the expression (for example, in terms of how many repeated terms there are). Nonetheless, this simplification procedure terminates, as it is impossible to get stuck in an infinite loop using the last operation alone. (Proof: we may assume that the $n_i$ are positive. Then all of the operations either reduce the number of terms, or the sum of the $n_i$.)

1. Is the proof above correct? (If not, and the original claim is correct, can you give me a representation of the number 5473?)

2. Assuming the proof is correct, is the original claim true for other non-integer bases? What I mean is the following:

The first three operations reduce the number of terms in the expansion and thus strictly simplify the expression (in terms of how many terms there are), but the last may need to be used several times before it "simplifies" the expression (for example, in terms of how many repeated terms there are). Nonetheless, this simplification procedure terminates, as it is impossible to get stuck in an infinite loop using the last operation alone. (Proof: we may assume that the $n_i$ are positive. Then all of the operations either reduce the number of terms, or leaves that unchanged and reduces the sum of the $n_i$.)

1. Is the proof above correct? (If not, and the original claim is correct, can you give me a representation of the number 5473?) Edit: Please see Michael Lugo's answer for a paper which finds the representation with the fewest nonzero digits in this "signed Fibonacci base". Please consider the following the actual question here:

2. Assuming the proof is correct, is the original claim true for other non-integer bases? What I mean is the following: