query: Prove that for all positive integers $n........$
positive_passages.docid: 155562
positive_passages.text: Proving $\sum_{k=1}^nk^3 = \left(\sum_{k=1}^n k\right)^2$ using complete induction
negative_passages.docid: 48081
negative_passages.text: Prove that for all $n \in \mathbb{N}$: $2^{n-1}(a^n + b^n) \ge (a + b)^n$
negative_passages.docid: 75166
negative_passages.text: Find with proof all positive integers $n$ such that $2^n + 1 \equiv 0 \pmod{n}$
negative_passages.docid: 20964
negative_passages.text: Proof that $n^2 < 2^n$
negative_passages.docid: 104920
negative_passages.text: Can we find positive integers $a$ and $k \geq 2$ with $2^n - 1 = a^k$?
negative_passages.docid: 20691
negative_passages.text: prove that $\frac{(2n)!}{(n!)^2}$ is even if $n$ is a positive integer
negative_passages.docid: 114404
negative_passages.text: For which positive integer values of $n$ does $a|mn-b?$
negative_passages.docid: 45645
negative_passages.text: Using permutation or otherwise, prove that $\frac{(n^2)!}{(n!)^n}$ is an integer,where $n$ is a positive integer.
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: 149758
negative_passages.text: How to prove $\log n < n$?
negative_passages.docid: 88144
negative_passages.text: Induction Proof $n! < n^n$
negative_passages.docid: 63661
negative_passages.text: integers $n$ such that $(n-1)!+1$ is a power of $n$
negative_passages.docid: 78726
negative_passages.text: Prove via mathematical induction that $4n < 2^n$ for all $ n≥5$.
negative_passages.docid: 27069
negative_passages.text: sum of all positive integers less than $n$ and relatively prime to $n$
negative_passages.docid: 62177
negative_passages.text: Prove that: $2^n < n!$ Using Induction
negative_passages.docid: 145891
negative_passages.text: Prove that $n! > \sqrt{n^n}, n \geq 3$
negative_passages.docid: 20438
negative_passages.text: Prove that every positive integer $n$ has a unique expression of the form: $2^{r}m$ where $r\ge 0$ and $m$ is an odd positive integer