query: For which integers is $i,j \geq 1$ exists an integer $k \geq 0$, such that $\frac{(i+j-1)!}{i!(j-1)!}=2^k$
positive_passages.docid: 143927
positive_passages.text: Binomial coefficients that are powers of 2
negative_passages.docid: 61850
negative_passages.text: Compute $\sum_{k=0}^{\infty}\frac{1}{2^{k!}}$
negative_passages.docid: 68400
negative_passages.text: Proof that $\frac{(2n)!}{2^n}$ is integer
negative_passages.docid: 66358
negative_passages.text: Prove by mathematical induction that $2n ≤ 2^n$, for all integer $n≥1$?
negative_passages.docid: 144836
negative_passages.text: How to prove that $\frac{(mn)!}{m!(n!)^m}$ is an integer?
negative_passages.docid: 146439
negative_passages.text: Is every sufficiently large positive integer of the form $ab + ac + bc + 1$?
negative_passages.docid: 91000
negative_passages.text: Prove that $\frac{1}{a+1}0$
negative_passages.docid: 81818
negative_passages.text: Prove or disprove : $\det(A^k + B^k) \geq 0$
negative_passages.docid: 151153
negative_passages.text: Prove $\frac{a(a^2+2)}{3}$ is an integer for all integer $a\geqslant 1$
negative_passages.docid: 65757
negative_passages.text: Prove that $n! + k$ is a composite number
negative_passages.docid: 107448
negative_passages.text: proving that there is no solution for $n$ in positive integers except $n = 1$ for $2^n = nk + 1$
negative_passages.docid: 36436
negative_passages.text: Proving $\sum_{k=1}^n k k!=(n+1)!-1$
negative_passages.docid: 14724
negative_passages.text: For primes $p≡3\pmod 4$, prove that $[(p−1)/2]!≡±1\pmod p$.
negative_passages.docid: 129801
negative_passages.text: Finding a closed form for $\sum^{n}_{k=1} \frac{k}{(k+1)!} $
negative_passages.docid: 14064
negative_passages.text: Proof by induction that $\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$
negative_passages.docid: 72600
negative_passages.text: Proof that $n!\leq {(\frac{n+1}{2})}^{n}$
negative_passages.docid: 46431
negative_passages.text: Proof that $\sum\limits_i \frac{(p-1)!}i$ is divisible by $p$