query: If $g(x+1)+g(x-1)=g(x)$ for all real $x$, then for what $p$ is $g(x+p)=g(x)$ necessarily true?
positive_passages.docid: 144222
positive_passages.text: If $f(x + 1) + f(x − 1) = f(x), \forall x \in \mathbb{R}$,then how to find $k$ such that $f(x + k) = f(x)$?
negative_passages.docid: 39843
negative_passages.text: Let $P$ be a prime. Show that $\exists$ $x \in \mathbb{N}$ such that $f(x) = p$ then $\exists$ $y \in \mathbb{N}$ such that $g(y) = p$
negative_passages.docid: 110313
negative_passages.text: Prove that $\gcd(f(x),g(x)) = 1$.
negative_passages.docid: 26207
negative_passages.text: If $f$ is a quadratic and $f(x)>0\;\forall x$, and $g= f + f' + f''$, prove $g(x)>0\; \forall x$
negative_passages.docid: 34738
negative_passages.text: Prove that $\gcd(n!+1,(n+1)!+1)=1$
negative_passages.docid: 120960
negative_passages.text: $f,g$ such that $\int fg = \int f \int g$
negative_passages.docid: 40383
negative_passages.text: Is it true that $|g| = |\phi(g)|$ for all homomorphisms $\phi: G \to G$ and $g \in G$?
negative_passages.docid: 2289
negative_passages.text: Prove that if $\gcd( a, b ) = 1$ then $\gcd( ac, b ) = \gcd( c, b ) $
negative_passages.docid: 145265
negative_passages.text: A non-zero function satisfying $g(x+y) = g(x)g(y)$ must be positive everywhere
negative_passages.docid: 73391
negative_passages.text: Prove that if $\gcd(a,b)=1$, then $\gcd(a^2,b^2)=1$
negative_passages.docid: 102777
negative_passages.text: Is $G^{(X, Y)} = (G^X)^Y?$ ($A^B$ just means that $B$ is mapped to $A$)
negative_passages.docid: 136782
negative_passages.text: Prove that $x * y = \frac{x+y}{1+xy}$ is a stable part of $G=(-1, 1)$
negative_passages.docid: 16035
negative_passages.text: Irreducibility of $X^{p-1} + \cdots + X+1$
negative_passages.docid: 63590
negative_passages.text: Let $g$ be a function from $(0, 1)$ to $\mathbb{R}$ and $\int_0^1 g^2(x)dx$ finite. Does this imply that $\int_0^1 g(x)dx$ is finite?
negative_passages.docid: 18714
negative_passages.text: $\gcd\left(a+b,\frac{a^p+b^p}{a+b}\right)=1$, or $p$
negative_passages.docid: 129605
negative_passages.text: $ 2g(z)=g(\frac{z}{2}) + g(\frac{z+1}{2})$
negative_passages.docid: 124742
negative_passages.text: P(A|C)=P(A|B)*P(B|C)?