query: Find all triples of non-negative Integers $a,b,c$ such that $a!b!=a!+b!+c!$
positive_passages.docid: 157746
positive_passages.text: Finding all positive integer solutions to $(x!)(y!) = x!+y!+z!$
negative_passages.docid: 17306
negative_passages.text: To find all $n$ such that $(n-1)!+1$ is a perfect power of $n$
negative_passages.docid: 70611
negative_passages.text: Prove by Mathematical Induction: $1(1!) + 2(2!) + \cdot \cdot \cdot +n(n!) = (n+1)!-1$
negative_passages.docid: 141102
negative_passages.text: If $a,b,c$ are positive integers, with $a^2+b^2-ab=c^2$ prove that $(a-b)(b-c)\le0$.
negative_passages.docid: 34299
negative_passages.text: Combinatorial proof of $\sum_{k=0}^n k \cdot k! = (n+1)! -1$
negative_passages.docid: 40343
negative_passages.text: Solutions of $p!q! = r!$
negative_passages.docid: 153763
negative_passages.text: Prove that $a_1!\cdots a_k! < n!$ whenever $a_1+\cdots+a_k < n$
negative_passages.docid: 20713
negative_passages.text: The sum of an uncountable number of positive numbers
negative_passages.docid: 126820
negative_passages.text: How many positive integer solutions are there to the equality $x_1+x_2+...+x_r= n$?
negative_passages.docid: 21000
negative_passages.text: Calculating $\sum_{k=1}^nk(k!)$ combinatorially
negative_passages.docid: 83862
negative_passages.text: How to prove that the number $1!+2!+3!+...+n!$ is never square?
negative_passages.docid: 135750
negative_passages.text: Prove that if a and b are integers, then there are unique integers q and r such that $a = bq + r$, $-|b|/2 < r \le |b|/2$
negative_passages.docid: 21531
negative_passages.text: The number of nonnegative integer triples with sum equal to $12$
negative_passages.docid: 33314
negative_passages.text: How many non-negative integer solutions does $x_1+x_2+\cdots+x_n=A$ have?
negative_passages.docid: 71012
negative_passages.text: If $a, b$ are positive integers such that $a|b^2, b^2|a^3,....$ then $a=b$.
negative_passages.docid: 75974
negative_passages.text: $\gcd(a,b)\!=\!1\!=\!\gcd(a,c)\Rightarrow\gcd(a,bc)\!=\!1$ [coprimes to $\,a\,$ are product closed]
negative_passages.docid: 66946
negative_passages.text: show that $(i+k)! > i!k!$ algebraically