query: Value of $a$ such that $ax^2-4x+9$ has integer roots
positive_passages.docid: 153647
positive_passages.text: Elementary number theory and quadratic
negative_passages.docid: 33469
negative_passages.text: Diophantine equation $ax + by = c$ has an integer solution $x_0, y_0$ if and only if $\gcd(a,b)|c$
negative_passages.docid: 33992
negative_passages.text: Let $\theta$ be a root of $p(x)=x^3+9x+6$, find the inverse of $1+\theta$ in $\mathbb{Q(\theta)}$
negative_passages.docid: 103246
negative_passages.text: Prove that a polynomial of degree $d$ has at most $d$ roots (without induction)
negative_passages.docid: 128107
negative_passages.text: find all $ a \in \mathbb N $ such that $2^a − 1 \equiv 0 \pmod 3$ and $ 4x^2 + 1 \equiv 0 \pmod {\frac{2^a -1}{3}}$ has a solution
negative_passages.docid: 148647
negative_passages.text: If $\dfrac{4x^2-1}{4x^2-y^2}$ is an integer, then it is $1$
negative_passages.docid: 79925
negative_passages.text: Infinitely many values of $x$ that satisfy the equations
negative_passages.docid: 20834
negative_passages.text: Integer solutions to $c=a^2-b^2$
negative_passages.docid: 6777
negative_passages.text: Showing $(a^{2}+2)(b^{2}+2)(c^{2}+2)\geq 9(ab+bc+ca)$
negative_passages.docid: 85653
negative_passages.text: $2^x + 3^x - 4^x + 6^x - 9^x ≤ 1$ $\forall x \in R$
negative_passages.docid: 5410
negative_passages.text: Prove that $a^2 + b^2 \geq 8$ if $ x^4 + ax^3 + 2x^2 + bx + 1 = 0 $ has at least one real root.
negative_passages.docid: 40107
negative_passages.text: primitive n-th roots of unity
negative_passages.docid: 98803
negative_passages.text: Square roots of integers and cyclotomic fields
negative_passages.docid: 141184
negative_passages.text: Proof that a degree 4 polynomial has at least two roots
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: 5981
negative_passages.text: For which values of $a$ does $a-2 \mid a^3+4$
negative_passages.docid: 70463
negative_passages.text: Let $p$ be prime and $a$ be a natural not divisible by $p$. Then, $\exists$ integers $x,y$ such that $ax\equiv y \pmod{p}$ with $0<|x|,|y|<\sqrt{p}$