query: Prove That $n(n+1)(n+2)(n+3)$ is never a square of a positive integer.
positive_passages.docid: 149716
positive_passages.text: Prove that $ \sqrt{n(n+1)(n+2)(n+3)+1} $ is always a natural number
negative_passages.docid: 113359
negative_passages.text: Proving that $x^2 + 4$ is not divisible by $3$
negative_passages.docid: 50894
negative_passages.text: Prove that $6$ divides $n(n + 1)(n + 2)$
negative_passages.docid: 4077
negative_passages.text: Using induction prove $n^3-n$ is divisible by 3 whenever n is a positive number.
negative_passages.docid: 108504
negative_passages.text: Prove that for all non-negative integers $m,n$, $\frac{(2m)!(2n)!}{m!n!(m + n)!}$ is an integer.
negative_passages.docid: 105405
negative_passages.text: Product of three consecutive positive integers is never a perfect power
negative_passages.docid: 79544
negative_passages.text: Prove $x^3 + 3 =4y(y+1)$ has no integer solutions
negative_passages.docid: 46102
negative_passages.text: Prove that $2^n +1$ in never a perfect cube
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: 96332
negative_passages.text: Prove by induction that $\sqrt[n+1]{n+1}<\sqrt[n]{n}$ for $n\ge 3$
negative_passages.docid: 77076
negative_passages.text: Proof that $(x-1)^2$ divides a $nx^{n+1} – (n+1)x^{n}+1$
negative_passages.docid: 127955
negative_passages.text: Show that $\sqrt{1+n(n+1)(n+2)(n+3)}$ is a whole number for all whole numbers $n$.
negative_passages.docid: 150088
negative_passages.text: Prove that $\sqrt {n − 1} +\sqrt {n + 1}$ is irrational for every integer $n ≥ 1$
negative_passages.docid: 51928
negative_passages.text: Use an induction argument to prove that for any natural number $n$, the interval $(n,n+1)$ does not contain any natural number.
negative_passages.docid: 4750
negative_passages.text: Show that for any positive integer n, $(3n)!/(3!)^n$ is an integer.
negative_passages.docid: 22973
negative_passages.text: Show that an integer of the form $8k + 7$ cannot be written as the sum of three squares.
negative_passages.docid: 56624
negative_passages.text: Prove that $0!+1! + 2! + 3! + ..... + n!$ $\neq$ $p^\text{r}$, where $n \geqslant 3$ and $n$, $p$ and $r$ are three integers