query: Prove that there'll always be three distinct integers $x$, $y$ and $z$ such that $x + y + z = 0$.
positive_passages.docid: 151375
positive_passages.text: There are $3$ Zero-Sum Numbers!
negative_passages.docid: 19571
negative_passages.text: Combinatorial proof for $\sum_{k = 0}^n \binom {r+k} k = \binom {r + n + 1} n$
negative_passages.docid: 90076
negative_passages.text: Prove that every positive integer $n$ is a unique product of a square and a squarefree number
negative_passages.docid: 133480
negative_passages.text: new addition and new multiplication x ⊕ y = x + y − 1, x ⊗ y = x + y − xy on set Z, prove the set Z equipped with these 2 new operation
negative_passages.docid: 30126
negative_passages.text: Linear Diophantine Equations in Three Variables
negative_passages.docid: 22367
negative_passages.text: How do I prove that for every positive integer $n$, there exist $n$ consecutive positive integers, each of which is composite?
negative_passages.docid: 28218
negative_passages.text: Elementary proof for $\sqrt[3]{xyz} \leq \dfrac{x+y+z}{3}$
negative_passages.docid: 125210
negative_passages.text: proof: primitive pythagorean triple, a or b has to be divisible by 3
negative_passages.docid: 86285
negative_passages.text: Given positive integers $ x , y$ and that $3x^2 + x = 4y^2 +y$ Prove that $x-y$ is a perfect square
negative_passages.docid: 103538
negative_passages.text: Partitions of $n$ into distinct odd and even parts proof
negative_passages.docid: 94067
negative_passages.text: How prove this diophantine equation $x^2+y^2+z^3=n$ always have integer solution
negative_passages.docid: 49427
negative_passages.text: $l^2+m^2=n^2$ $\implies$ $lm$ is always a multiple of 3 when $l,m,n,$ are positive integers.
negative_passages.docid: 85525
negative_passages.text: Prove that $x^3+x+1$ splits in $Z_2[x]/(x^3+x+1)$
negative_passages.docid: 92290
negative_passages.text: For each $x,y,z∈\mathbb N$, if $x<y$ then $x+z<y+z$
negative_passages.docid: 33786
negative_passages.text: Proof that a certain fraction is always an integer
negative_passages.docid: 70662
negative_passages.text: How many non-negative integer solutions does the equation $3x + y + z = 24$ have?
negative_passages.docid: 148243
negative_passages.text: Prove that there are no positive integers $x$ and $y$ such that $x^3 + y^3 = 10^3$.