query: Prove there are no positive integers $x$ and $y$ such that $x^3 + y^3 = 10^3$.
positive_passages.docid: 148243
positive_passages.text: Prove that there are no positive integers $x$ and $y$ such that $x^3 + y^3 = 10^3$.
negative_passages.docid: 41364
negative_passages.text: How to prove that $x^y - y^x = x + y$ has only one solution in positive integers?
negative_passages.docid: 34767
negative_passages.text: Proving $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)<3$ for all positive integers $n$
negative_passages.docid: 51649
negative_passages.text: prove there is no smallest positive rational number
negative_passages.docid: 39346
negative_passages.text: Let $R= \mathbb{Q}[x^2 ,x^3 ]$, the set of all polynomials over $\mathbb{Q}$ with no $x$ term. Show that the $\gcd$ of $x^5$ and $x^6$ does not exist
negative_passages.docid: 113268
negative_passages.text: Proving $x^4+y^4=z^2$ has no integer solutions
negative_passages.docid: 4750
negative_passages.text: Show that for any positive integer n, $(3n)!/(3!)^n$ is an integer.
negative_passages.docid: 96153
negative_passages.text: $ x^3<y^3 \iff x<y $ proof issue.
negative_passages.docid: 15753
negative_passages.text: Solve the following equation in positive integers $x$ and $y$
negative_passages.docid: 104498
negative_passages.text: Show that $x^3y^3(x^3+y^3) \leq 2$
negative_passages.docid: 86378
negative_passages.text: Prove that $3(a^3+b^3+c^3) > (a+b+c)(a^2+b^2+c^2)$
negative_passages.docid: 134121
negative_passages.text: Determining all the positive integers $n$ such that $n^4+n^3+n^2+n+1$ is a perfect square.
negative_passages.docid: 137010
negative_passages.text: What are integers $x$ and $n$ such as $ x^4+x^3+x^2+x+1 = n^2$?
negative_passages.docid: 99055
negative_passages.text: How find this $x^3-5x+10=2^y$
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: 113359
negative_passages.text: Proving that $x^2 + 4$ is not divisible by $3$
negative_passages.docid: 69902
negative_passages.text: $3$ never divides $n^2+1$