Zongxiang Yi
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If $a \equiv b \pmod n$ and $c+d = n$, does $ca+bd \equiv 0 \pmod n$?
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4 votes

$a\equiv b\pmod{9}$ and $c+d=9$ implies $a-b\equiv 0\pmod{9}$ and $d\equiv -c \pmod{9}$ ,respectively. So $$ca+bd\equiv ca-cb\equiv c(a-b)\equiv c\cdot 0\equiv 0 \pmod{9}.$$

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Prove that there are infinitely many integers $n>0$ such that $n^2+n+1$ divides $4^n+2^n+1$
4 votes

ans:=[n: n in [2^(2^k):k in [0..11]] | (Modexp(4,n,n^2+n+1)+Modexp(2,n,n^2+n+1)+1) mod (n^2+n+1) eq 0 ]; print #ans, ans; After running the above magma script, it outputs 12 [ 2, 4, 16, 256, 65536, ...

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Representation of Euler phi function
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3 votes

Theorem: $\varphi$ cannot be represented by $\sum_{d|n, d>0}f(d)$ for some funtion $f$. Proof: Assume that there exists a function $f$, such that $\varphi(n)=\sum_{d|n, d>0}f(d)$. Consider some ...

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Proofs involving strict inequalities
3 votes

Let $b=f(a)=2+\sqrt{a-2}$, where $a\in (3,5)$. You have proved that $3<f(a)=b$. Now consider $$g(a)=a-f(a)=a-2-\sqrt{a-2}.$$ Then $g(a)$ is a increasing function since $g'(a)=1-\frac{1}{2\sqrt{a-2}}...

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Find the period of $f(x)$ with $f(x)f(y)=f(x+y)+f(x-y)$.
3 votes

To summary, I am going to show the following result. Main Theorem: Let $f$ be the function mentioned in OP. If the peroid of $f$ exists, then the peroid of $f$ only has one of the two forms: $\...

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What is the difference between codomain and range?
2 votes

Codomain is a set which the images must belong to. Range is the set which the images exactly belongs to.

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$R$ finite commutative ring with identity. Prove that $a \in R$ is either a zero divisor or a unit.
1 votes

Let $a \in R$. We must prove that $a$ is a zero divisor or a unit. Consider the map $\varphi_a(r)=ar$ from $R$ to $R$. One Proof If $a$ is a unit, then $\varphi_a$ is a bijection. Assume that $a$ ...

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Problem regarding open set/closed set
1 votes

Proposition: In $\mathbb{R}^n$, if $U$ is an open set and $V$ is a closed set, then $U+V$ is an open set. Proof: Note that for any point $z\in U+V$, there exist $x\in U$ and $y \in V$ such that $z=...

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Property of modular arithmetic. $a \bmod p = b \bmod p \iff p \uparrow |a-b|$
1 votes

I think the binary operation of "mod" in the OP refers to the remainder. Let $a=q_1 p + r_1$ and $b=q_2 p + r_2$, where $0\le r_1,r_2<p$. Then $$r_1=(a \bmod p),\ r_2=(b\bmod p).$$ If $\,a \bmod ...

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without actually finding them, determine the number of solutions of the congruence.
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1 votes

Here is another solution which follows the direction of your professor. You can recall some thing in your another post here. We use the same notations in that post. 1) Let $n=11^2\cdot 23^2$. It is ...

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without actually finding them, determine the number of solutions of the congruence.
1 votes

Let $S(f,n)$ denote the set of solutions of congruence equation $f(x)\equiv 0 \pmod{n}$ for $f=\sum_{i=0}^{m}{a_ix^i}\in\mathbb{Z}[x]$. Denote $N(f,n)=|S(f,n)|$. Lemma 1: Let $n=\prod_{i=1}^{k}{...

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Primes having 2 as a primitive root
1 votes

As the comments point out whether $2$ is a primitive root of $p$ depends on $p$, $a$ and $y$, this answer is trying to figure out WHEN $2$ is a primitive root of $p$. As $$a\equiv 2^y \pmod{p},$$ ...

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Prove $ P(A-B)=P(A)-P(A\cap B) $
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1 votes

Due to the definition of events, we can treat events as sets. https://en.wikipedia.org/wiki/Event_(probability_theory) Lemma: $A-B=A-(A\cap B)$. Proof:According to the definition of the difference ...

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Prove that $inf(S_1)=sup(S_2)$
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1 votes

Lemma 1. If $f$ is continuous on $[a,b]$, then for any $y_0 \in (N,M)$ there exists a point $x_0 \in (a,b)$ such that $f(x_0)=y_0$, where $N=\min(f(a),f(b))$ and $M=\max(f(a),f(b))$. Since $f$ is ...

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what is the absolute value of a set?
1 votes

In particular, there are only two equivalence classes of $S$. One is, denoted by $\overline{0}$, $$\{ \emptyset, \{1, 2\}, \{ 1, 3\}, \{1, 4\}, \{2,3\},\{2,4\},\{3,4\},\{1,2,3,4\} \}.$$ And the other ...

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How to prove A → (B ∨ C) given A → B
1 votes

Use truth table. A B C A->B A->(BVC) 0 0 0 1 1 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 1 0 1 0 1 1 1 0 1 1 1 1 1 1 1 You can see that when A $\...

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Proving that $g(x)$ defined as $x^2$ on the rationals and $x^4$ on irrationals, is discontinuous at $2$
1 votes

I think a simpe way is to construct two sequences $a_n$ and $b_n$, where $a_n$ are rational and $b_n$ are irrational, such that $\lim_{n\to\infty}g(a_n)=2^2$ and $\lim_{n\to\infty}g(b_n)=2^4$. Then ...

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Find the lcm of $23,24,30,32,40,41,43,50$ where these numbers are in base $6$
1 votes

Note that $lcm(a,b,c)=lcm(lcm(a,b),c)$ and $lcm(a,b)=\frac{ab}{\gcd(a,b)}$, so what you should do the to understand the multiplication and division (https://en.wikipedia.org/wiki/Euclidean_division )...

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how does one solve $\lim_{n\to \infty} (\frac{n^{2}-2n+1}{n^{2}-4n+2})^{n}$
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1 votes

Lemma: If $g(n)>0$, $h(n)>0$ when $n \to \infty$ and $$\lim_{n \to \infty}g(n)=a, \lim_{n \to \infty}h(n)=b,$$ then $$\lim_{n \to \infty}g(n)^{h(n)}=a^b.$$ So $$\lim_{n \to \infty}(\frac{n^2-...

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Show that a group of order 2016 has a subgroup of order 28
1 votes

By Sylow's theorems, there exists a sylow 2-subgroup $G_1$ and a sylow 7-subgroup $G_2$. Note any p-subgroups $G$ have non-trival center $Z(G)$. Now you can choose $a \in Z(G_1)$ and $b \in Z(G_2)$ ...

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prove that $x_{n+1}=c+\sqrt{x_n}$
1 votes

So you have $x_{n+1}-x_n=c+\sqrt{x_n}-x_n$. Now consider the function: $$f(x)=c+\sqrt{x}-x,x\in R.$$ It follows $$f'(x)=\frac{1}{2\sqrt{x}}-1.$$ You can see that when $x< \frac{1}{4}$, it has $$f'(...

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Question about the proof of "If $H$ is a Sylow $p$-subgroup of $G$, then $|H|=p^n$.
0 votes

What is "maximal" ? In general, let $(U,\le)$ be a partially ordered set. An element $m\in U$ is called maximal if $x\le m$ for all $x\in \{u\in U \mid u\le m \text{ or } m\le u\}$. In OP, $...

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Finding solutions of modulus functions
0 votes

Note that $|x|=\sqrt{x^2}$. So $$|x-5|=|3x-2|\Leftrightarrow \sqrt{(x-5)^2}=\sqrt{(3x-2)^2}.$$ Lemma:If $x,y>0$, then $x=y$ if and only if $\sqrt{x}=\sqrt{y}$. Since $(x-5)^2>0$ and $(3x-2)^...

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Let $A=\begin{bmatrix} 1 & 2\\ 3& 4 \end{bmatrix}$ then det$(A^3-6A^2+5A+3I)=3$
0 votes

Here is another method. Assume that $\lambda_i\in\mathbb{C}$, $i=1,2$ are the two igenvalues of $A$. Let $f(x)\in \mathbb{C}[x]$ be the characteristic polynomial of $A$. Then $$f(x)=\det(A-xI)=x^2-5x-...

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Is $e^x$ the only non-trivial function for which the differential operator is the identity operator?
0 votes

If consider functions of the form $ke^x$, then YES. Assume $f(x)>0$. As $$\mathrm{\frac{d(\log(f(x)))}{dx}}=\frac{1}{f(x)}\cdot\mathrm{\frac{df(x)}{dx}}=1,$$ We have $$\int{\mathrm{\frac{d(\log(f(...

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Prove $\forall m\in\mathbb{N},m\neq1:\quad\sum_{n=1}^{m}\frac{1}{n^2}\leq\int_1^m\frac{\sqrt{x^6+4}}{x^3}\ dx$
0 votes

@Hagen von Eitzen shows that it holds for $m\ge 3$. Still it need to check if $$\int_1^2\frac{\sqrt{x^6+4}}{x^3}\,\mathrm dx\ge\frac54.$$ This answer will prove that it holds for $m=2$. Let $f(x)=\...

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How many solutions the equation $x^3=(1\space 2\space 3\space 4)$ have in $S_7$
0 votes

For simplicity, denote $[n]=\{1,2,\cdots,n\}$. Lemma 1: Let $S$ be an infinite set and $\sigma:S \to S$ be a finitary permutation -- a permutation that moves only finitely many elements. Then, ...

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Inverse of $α∈Z_3(α)$ where $α^3+α^2+2=0$
0 votes

For any element $x\in \mathrm{GF}(3^3)$, there exist $a_i\in \mathbb{Z}_3$, $i=0,1,2$, such that $x=\sum_{i=0}^{2}a_i\alpha^i$. Suppose $y=\sum_{i=0}^{2}b_i\alpha^i$ such that $xy=1$, where $b_i\in \...

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List all the possible sizes of N. Which of these answers divides 60?
0 votes

A:=1; Bs:=[*12,12,15,20*]; SS:=Exclude(Subsets({i : i in [1..#Bs]}), {}); N:={<[Bs[i]: i in s], &+[Bs[i]: i in s] + A> : s in SS}; print {s[2] : s in N}; print {s : s in N | s[2] mod 60 eq 0}...

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Solving equation with infinite exponent tower
0 votes

The key point is to notice that $\lim_{n\to \infty}{a_n}=\lim_{n\to \infty}{a_{n+k}}$ for any $k\in \mathbb{Z}$. It means that the limitation does not depend on the first $k$ items. Define a sequence:...

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