query: Find all pairs of integers $(n,k)$ that satisfy $(n+1)^k - 1 = n!$
positive_passages.docid: 17306
positive_passages.text: To find all $n$ such that $(n-1)!+1$ is a perfect power of $n$
negative_passages.docid: 34299
negative_passages.text: Combinatorial proof of $\sum_{k=0}^n k \cdot k! = (n+1)! -1$
negative_passages.docid: 30261
negative_passages.text: How can I find all integers $x≠3$ such that $x−3|x^3−3$
negative_passages.docid: 47799
negative_passages.text: Show that the number or $r$ combinations of $X$ which contain no consecutive integers is given by $\binom{n-r+1}{r}$
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: 41354
negative_passages.text: Binomial-coefficients if, k, m, n natural numbers and k \leq n the result of
negative_passages.docid: 96861
negative_passages.text: How find all ordered pairs $(m,n)$ such $\frac{n^3-1}{mn-1}$ is an integer
negative_passages.docid: 63383
negative_passages.text: Integers $n$ for which $|2^n + 5^n – 65|$ is a perfect square
negative_passages.docid: 12967
negative_passages.text: Find all $n>1$ such that $\dfrac{2^n+1}{n^2}$ is an integer.
negative_passages.docid: 155363
negative_passages.text: Find all integers $(x, y)$ such that $1 + 2^x + 2^{2x + 1} = y^2$
negative_passages.docid: 9217
negative_passages.text: Find $n$ such that $x^2 + x + 1$ is a factor of $(x+1)^n - x^n - 1$.
negative_passages.docid: 8034
negative_passages.text: Can $n!$ be a perfect square when $n$ is an integer greater than $1$?
negative_passages.docid: 8884
negative_passages.text: Find all natural numbers $n$ and $m$ for which $n^m=(n-1)!+1$
negative_passages.docid: 98443
negative_passages.text: Find positive integers such that $2n^3 + 5|n^4 +n+1$
negative_passages.docid: 36436
negative_passages.text: Proving $\sum_{k=1}^n k k!=(n+1)!-1$
negative_passages.docid: 98082
negative_passages.text: Another combinatorics problem: $\sum\limits_{k = 0}^n (-1)^k \binom{2n-k}k2^{2n-2k}=2n+1$
negative_passages.docid: 83738
negative_passages.text: Prove $(k,n+k)=1$ iff $(k,n)=1$