query: How to prove that $(X^y - X) \bmod 3 = 0$ always when $y$ is odd and $y > 1$, and $X >1$?
positive_passages.docid: 49938
positive_passages.text: Proof of $a^{\phi (n) +1} \equiv a \pmod{\! n} $ Carmichael generalization of Fermat-Euler's theorem.
negative_passages.docid: 66451
negative_passages.text: In a ring, how do we prove that a * 0 = 0?
negative_passages.docid: 20908
negative_passages.text: How to compute $k$'th roots $\!\bmod n$?
negative_passages.docid: 10105
negative_passages.text: Proving $\, a^b = (a\bmod n)^b\, \pmod{\!n}\ $ [Congruence Power Rule]
negative_passages.docid: 81346
negative_passages.text: Let $r$ be the smallest positive integer such that $a^r \equiv 1 \bmod{n}$. Prove $r \mid \phi(n)$.
negative_passages.docid: 155757
negative_passages.text: Number theory: $x^y + 1 = y^x$
negative_passages.docid: 41364
negative_passages.text: How to prove that $x^y - y^x = x + y$ has only one solution in positive integers?
negative_passages.docid: 33332
negative_passages.text: A classic exponential inequality: $x^y+y^x>1$
negative_passages.docid: 90438
negative_passages.text: To prove that if $ab\equiv ac\bmod n$ and $(a,n)=1$ then $b\equiv c\bmod n$
negative_passages.docid: 92537
negative_passages.text: Prove that $f(x)\in \mathbb{Z}[x]$ such that $f(0)$ and $f(1)$ are odd has no integer roots
negative_passages.docid: 7414
negative_passages.text: Prove that odd perfect square is congruent to $1$ modulo $8$
negative_passages.docid: 27281
negative_passages.text: How to prove $\,x^a-1 \mid x^b-1 \iff a\mid b$
negative_passages.docid: 142724
negative_passages.text: Prove that for all integers $x$ and $y$, $x - y$ is odd if and only if $x + y$ is odd.
negative_passages.docid: 20626
negative_passages.text: Prove that $x^y < y^x$
negative_passages.docid: 34693
negative_passages.text: Prove $E^{\mathbb Q}[Y]=E^{\mathbb P}[XY]$ if $E^{\mathbb P}[X]=\mathbb P(X>0)=1$ and $ \mathbb Q(A)=E^{\mathbb Q}[X1_A] $
negative_passages.docid: 139659
negative_passages.text: Find all $n$ with $\!\bmod n\!:\, a\,$ invertible $\Rightarrow a^{-1} \equiv a,\,$ i.e. $a^2\equiv 1$
negative_passages.docid: 14677
negative_passages.text: Proving the exponential inequality: $x^y+y^x\gt1$