query: prove that there are infinitely many primitive pythagorean triples $x,y,z$ such that $y=x+1$
positive_passages.docid: 28781
positive_passages.text: $k^{2}+(k+1)^{2}$ being a perfect square for infinitely many $k$
negative_passages.docid: 54430
negative_passages.text: For $a,b$ coprime, there exists positive integers $x,y$ such that $ax-by=1$
negative_passages.docid: 83496
negative_passages.text: Math contest integer triplet problem: $x + y + z + xy + yz + xz = xyz − 1$
negative_passages.docid: 75957
negative_passages.text: Prove $x^3+3y^3+9z^3-9xyz=1$ has infinity integers solutions!
negative_passages.docid: 7279
negative_passages.text: $|2^x-3^y|=1$ has only three natural pairs as solutions
negative_passages.docid: 47162
negative_passages.text: Proving there are infinitely many primes of the form $a2^k+1.$
negative_passages.docid: 124223
negative_passages.text: Find all pairs (x, y) of real numbers such that $16^{x^2+y} + 16^{x+y^2}=1$
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: 84021
negative_passages.text: How can we show that $\varphi(a) \neq a +1$ for every $a \in A$?
negative_passages.docid: 49209
negative_passages.text: Prove $(x+y)(y+z)(x+z)≤\frac8{27}(x+y+z)^3$
negative_passages.docid: 150528
negative_passages.text: Pythagorean triples $(a,b,c)$ and divisibility of $a$ or $b$ by $3$.
negative_passages.docid: 16176
negative_passages.text: Primitive Pythagorean triple divisible by 3
negative_passages.docid: 24954
negative_passages.text: Prove that there are infinitely many integers of the form $111\ldots111$ that are multiple of $n$.
negative_passages.docid: 145287
negative_passages.text: Solve for integers $x, y, z$ such that $x + y = 1 - z$ and $x^3 + y^3 = 1 - z^2$.
negative_passages.docid: 142835
negative_passages.text: Proving :$\arctan(1)+\arctan(2)+ \arctan(3)=\pi$
negative_passages.docid: 55673
negative_passages.text: Alternate proof that $\exists\ x \in \Bbb{Z}$ such that $ \gcd (a+bx,c) = 1$?
negative_passages.docid: 65000
negative_passages.text: prove that $x^2 + y^2 = z^4$ has infinitely many solutions with $(x,y,z)=1$