query: proof - any positive integer can be uniquely expressed as $a = 3^m + 3^{m-1}b_{m-1} + \cdots + 3^0b^0$
positive_passages.docid: 150140
positive_passages.text: Show that every nonzero integer has balanced ternary expansion?
negative_passages.docid: 114074
negative_passages.text: Intuitive/Visual proof that $(1+2+\cdots+n)^2=1^3+2^3+\cdots+n^3$
negative_passages.docid: 116136
negative_passages.text: Proof that the inverse of a square matrix is unique
negative_passages.docid: 29692
negative_passages.text: Proof verification: elementary Number Theory Proof of $am+bn = 1$
negative_passages.docid: 128575
negative_passages.text: Elementary proof that $3$ is a primitive root of a Fermat prime?
negative_passages.docid: 56138
negative_passages.text: Proving $\sum_{k=0}^{2m}(-1)^k{\binom{2m}{k}}^3=(-1)^m\binom{2m}{m}\binom{3m}{m}$ (Dixon's identity)
negative_passages.docid: 140140
negative_passages.text: An identity which applies to all of the natural numbers
negative_passages.docid: 56528
negative_passages.text: On the proof of existence and uniqueness of radix representation
negative_passages.docid: 144592
negative_passages.text: how to prove the existence and uniqueness of the prime factorization of any natural number.
negative_passages.docid: 30136
negative_passages.text: Proof that a Combination is an integer
negative_passages.docid: 145397
negative_passages.text: How to prove that for every $i$, $a_i = i$
negative_passages.docid: 22367
negative_passages.text: How do I prove that for every positive integer $n$, there exist $n$ consecutive positive integers, each of which is composite?
negative_passages.docid: 146439
negative_passages.text: Is every sufficiently large positive integer of the form $ab + ac + bc + 1$?
negative_passages.docid: 119046
negative_passages.text: Expressing a positive integer as a sum of positive integers
negative_passages.docid: 127468
negative_passages.text: Prove that there is only one unique base b representation of any natural number.
negative_passages.docid: 114404
negative_passages.text: For which positive integer values of $n$ does $a|mn-b?$
negative_passages.docid: 111406
negative_passages.text: Proving that a definition of e is unique