query: Every natural number can be expressed uniquely as the sum of $k$ binomial coefficients
positive_passages.docid: 152845
positive_passages.text: Every integer is the unique sum of a "decreasing" sequence of binomial coefficients
negative_passages.docid: 65757
negative_passages.text: Prove that $n! + k$ is a composite number
negative_passages.docid: 48308
negative_passages.text: Uniqueness of radix representation of integers in base 3
negative_passages.docid: 7083
negative_passages.text: Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$
negative_passages.docid: 75910
negative_passages.text: Combinatorial Proof of Combinatorial Identity involving $(-1)^k \binom {n-1}{k}$
negative_passages.docid: 142534
negative_passages.text: Proof of $k {n\choose k} = n {n-1 \choose k-1}$ using direct proof
negative_passages.docid: 105237
negative_passages.text: Alternating sum of binomial coefficients identity
negative_passages.docid: 67927
negative_passages.text: Combinatorial argument for the identity $k\binom{n}{k} = n\binom{n-1}{k-1}$
negative_passages.docid: 33104
negative_passages.text: Combinatorial Proof with Summation Identity
negative_passages.docid: 56985
negative_passages.text: Show that every positive integer has a unique representation in the form $50k+l$...?
negative_passages.docid: 142584
negative_passages.text: Confusion in proof that primes $p = 4k + 1$ are uniquely the sums of two squares
negative_passages.docid: 45392
negative_passages.text: Every natural number can be written as the sum of distinct Fibonacci numbers?
negative_passages.docid: 107021
negative_passages.text: Finding a combinatorial argument for an interesting identity: $\sum_k \binom nk \binom{m+k}n = \sum_i \binom ni \binom mi 2^i$
negative_passages.docid: 32608
negative_passages.text: Binary expansion Unique
negative_passages.docid: 108401
negative_passages.text: Sum of binomial coefficients $\sum_{k=0}^{n}(-1)^k\binom{n}{k}\binom{2n - 2k}{n - 1} = 0$
negative_passages.docid: 23707
negative_passages.text: How can I prove that every word has unique representation?
negative_passages.docid: 87320
negative_passages.text: Proving identities like $\sum_{k=1}^nk{n\choose k}^2=n{2n-1\choose n}$ combinatorially