query: Prove that there do not exist distinct integers $a,b,c$ and polynomial $P$, with integer coefficients, such that $P(a)=b, P(b)=c, P(c)=a$.
positive_passages.docid: 1962
positive_passages.text: Polynomial $P(a)=b,P(b)=c,P(c)=a$
negative_passages.docid: 95111
negative_passages.text: Elementary proof for $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ where $p_i$ are different prime numbers.
negative_passages.docid: 19016
negative_passages.text: Given 3 distinct primes {$p,q,r$}, then $|G|=pqr \implies G$ not simple
negative_passages.docid: 27540
negative_passages.text: A polynomial can not have infinite number of constant values
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 $\sqrt{a}=b\sqrt{c}$.
negative_passages.docid: 94682
negative_passages.text: Polynomial: prove there is no double root
negative_passages.docid: 102980
negative_passages.text: finding all numbers $a,b$ that divide $c$ but product of $a$ and $b$ does not divide $c$
negative_passages.docid: 134857
negative_passages.text: Find $(a,b)$ if $x^2-bx+a = 0, x^2-ax+b = 0$ both have distinct positive integers roots
negative_passages.docid: 20003
negative_passages.text: Prove for positive integers a,b,c and d (where b does not equal d), if gcd(a,b) = gcd(c,d) = 1, then a/b + c/d is not an integer
negative_passages.docid: 67891
negative_passages.text: Why are there no discrete zero sets of a polynomial in two complex variables?
negative_passages.docid: 77806
negative_passages.text: Proving if $a$ is not a multiple of prime $p$, then an integer $b$ exists such that $p^b - 1$ is a multiple of $a$
negative_passages.docid: 11564
negative_passages.text: $\mathbb{Q}(\sqrt{p},\sqrt[3]{q})=\mathbb{Q}(\sqrt{p}\sqrt[3]{q})$ for distinct prime $p,q$
negative_passages.docid: 99000
negative_passages.text: No polynomial with integer coefficients satisfies the conditions
negative_passages.docid: 20881
negative_passages.text: Prove that if $p$ and $q$ are distinct primes such that $pq\mid n^2$ then $pq\mid n$.
negative_passages.docid: 51094
negative_passages.text: Prove that the set of integer coefficients polynomials is countable
negative_passages.docid: 131289
negative_passages.text: Polynomial with a root modulo every prime but not in $\mathbb{Q}$.
negative_passages.docid: 71110
negative_passages.text: Show that we cannot have a prime triplet of the form $p$, $p + 2$, $p + 4$ for $p >3$