query: All $n$ such that $n+2$ is a cube and $n-2$ is a square.
positive_passages.docid: 22112
positive_passages.text: Find all integer solutions to $x^2+4=y^3$.
negative_passages.docid: 41789
negative_passages.text: Fermat's theorem on sums of two squares
negative_passages.docid: 92057
negative_passages.text: Prove that for every odd integer n, there is a group...
negative_passages.docid: 4194
negative_passages.text: Which integers can be expressed as a sum of squares of two coprime integers?
negative_passages.docid: 93657
negative_passages.text: Proving “The sum of $n$ consecutive cubes is equal to the square of the sum of the first $n$ numbers.”
negative_passages.docid: 17047
negative_passages.text: Prove that if n is even then $n^2$ is even and if n is odd then $n^2$ is odd?
negative_passages.docid: 71935
negative_passages.text: Let $k$ be a natural number . Then $3k+1$ , $4k+1$ and $6k+1$ cannot all be square numbers.
negative_passages.docid: 22800
negative_passages.text: Can $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m$ are perfect squares?
negative_passages.docid: 155784
negative_passages.text: If $\gcd(a, b) = 1$ and if $ab = x^2$, prove that $a, b$ must also be perfect squares; where $a,b,x$ are in the set of natural numbers
negative_passages.docid: 134000
negative_passages.text: Proof by Strong Induction: $n = 2^a b,\, b\,$ odd, every natural a product of an odd and a power of 2
negative_passages.docid: 96185
negative_passages.text: Solutions of $x^2+y^2+z^2 = n$ in $\mathbb{Z}_2$.
negative_passages.docid: 3479
negative_passages.text: How to count the no. of distinct ways in which $1,2,...,6$ can be assigned to $6$ faces of a cube?
negative_passages.docid: 98440
negative_passages.text: Show that any non-negative rational integer may be written as the sum of two squares if it is of the form....
negative_passages.docid: 20964
negative_passages.text: Proof that $n^2 < 2^n$
negative_passages.docid: 142580
negative_passages.text: Prove that a perfect square is either a multiple of $4$ or of the form $4q+1$.
negative_passages.docid: 136964
negative_passages.text: Prove there exist 135 consecutive integers that each have a perfect cube $>1$ that divides it.
negative_passages.docid: 31436
negative_passages.text: Show that for all $a\in\mathbb{N}$, there exists $b\in\mathbb{N}$ and square-free integer $c$ such that $\sqrt{a}=b\sqrt{c}$.