Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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one-to-one correspondence with a set of primitive dirichlet charchter

Let $$\operatorname{Prim}_{N}=\{\xi \mid \xi \text{ a primitive Dirichlet charchter mod } F \text{ with } F\mid N\}$$ and $$\operatorname{Char}_{N}=\{\xi" \mid \xi" \text{ Dirichlet charchter mod } N ...
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2answers
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quadratic diophantine expression

Can someone suggest a reference? Also, why are there 7 answers to the question 39*c^2 + 3*c + 1 mod 49 is congruent to 0 see my Maple worksheet. I sort of answered my own question. If 49 ...
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0answers
21 views

how to predict the sum of digits of the number A(n) for a large natural n without calculation , A(n)=a(n²+n)+b?

look A(n)=9n²+9n-1 , let n=15233 , A(15233)=2088535697 the sum of digits of this obtained number is :53 and always take this form :9k+8 , where k=5 and always exist a natural number k give us the ...
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2answers
15 views

Solve for x when x is on both sides of modular equation

This question is purely out of curiosity. My little brother got a question for homework to find a rectangle where the Area = Outline. Both sides must also be integers, obviously. He found the square ...
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0answers
12 views

Previous step of the supplement to the law of cubic reciprocity

Let $\gamma$, $\rho\in\mathbb{Z[\omega]}$ be different primary irreducibles (i.e. $\gamma$, $ \rho\equiv 2(3)$), where $\omega=-\frac{1}{2}+i\frac{\sqrt{3}}{2}$. I have to prove that ...
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0answers
5 views

Solution of Pell equation over field of p-adic numbers

Right now I am studying Pell equation. Using continued fractions, we can find the solution of Pell equation. Now my question, is it possible for me to find a solution in the field of p-adic numbers ...
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0answers
18 views

Which theorem could be used?

I want to write the $p-$adic expansion of $6!$ in $\mathbb{Q}_3$. I have to solve the congruence $x \equiv 6! \pmod {3^n}$, right? I found the following: $$x_0 \equiv 6! \pmod 3 \Rightarrow x_0 ...
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21 views

Showing two elements are coprime in a ring of integers

Let $\alpha$ and $\beta$ be the two roots of the polynomial $x^2 - x + 2$. I was wondering if someone could explain to me why $(y - \alpha)$ and $(y - \beta)$ are coprime (for any integer $y$) in the ...
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1answer
24 views

Prove the Inequality on Prime Counting Function

Is there any way to prove that, $$\pi(x^2)-\pi(y^2) \geq \sqrt{\pi(x-y)}$$ I have tried to prove it using inequalities on $\pi(x)$ but it didn't work. Can anyone help me?
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4answers
43 views

Prove that $x^2 + y^2 = 3(z^2 + m^2)$ has no solutions in integer [on hold]

Prove that: $$ x^2 + y^2 = 3(z^2 + m^2) $$ has no solutions in integer Except $0 0 0 0$
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1answer
58 views

Integral solutions of $x^5-27y^3=2x$

Find all integers $x$ and $y$ such that $x^5-27y^3=2x$.
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1answer
31 views

Given $0 < p < n$, prove there exists $n$ consecutive natural numbers such that each natural is divisible by at least $p$ distinct primes.

Given $0 < p < n$, prove there exists $n$ consecutive natural numbers such that each natural is divisible by at least $p$ distinct primes. Is there a general proof method to prove this ...
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2answers
65 views

Two math professors problem

My friend asks me a question from internet. The question is as follows Two math professors, professor Uno and professor Dos, play chess at the park while reminiscing about their past. Prof. ...
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2answers
40 views

Find all natural numbers n for which $3^n + 5^n$ is divisible by $3^{n-1} + 5^{n-1}$

Find all natural numbers n for which $3^n + 5^n$ is divisible by $3^{n-1} + 5^{n-1}$. Interested if there is a nice quick way other than mine.
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5answers
45 views

How to know a number is divisible by a given number without using a calculator?

My question is simple and comes from my curiousity during studying math. How to know a number is divisible by $7$ or $13$ without using a calculator? For example, how do we decide intuitively that ...
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0answers
27 views

What does “combining the solutions in O(n) time” mean?

Algorithm $X$ proceeds by recursively solving $5$ subproblems of one-half the size, then combining the solutions in $O(n\log n)$ time. Algorithm $Y$ makes $9$ recursively calls on ...
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1answer
32 views

How to quickly determine running time of such recurrence relations?

$$T(n)=5T(\frac{n}{2})+n\log n$$ $$T(n)=9T(\frac{n}{3})+n^2$$ $$T(n)=2T(\frac{2n}{3})+n^{1.5}$$ What are the running times of each $T(n)$? Each one looks like the form of the Master Theorem, but only ...
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2answers
27 views

Solve the recurrence $T(n)=3T(n/3)+\log n$, $T(1)=1$

So $T(n)=3T(n/3)+\log n$ and $T(1)=1$. I tried to solve this by expanding it out to see a pattern, but I don't really see the pattern: $T(n/3) = 3T(n/9)+\log (n/3)$ $T(n) = 3[3T(n/9)+\log ...
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3answers
58 views

Finding all the values of n, such that $ \varphi (n) = 12 $ [duplicate]

I have not broken this down very far. I have come to the conclusion that there are infinitely many values for n where there exists 12 coprimes to n. Since there are infinitely many primes, and primes ...
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2answers
14 views

A problem regarding Extended Euclidean Algorithm

A Linear Diophantine Equation is of the following form: Ax+By+C=0, where,gcd(A,B)=d and A=da,B=db.If (x1,y1) is a solution of the diophantine equation, every solution is of the form: x=x1+bt,y=y1−at ...
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0answers
48 views

What is known about$\sum\limits_{p\text{ prime}} \frac{1}{p^2-1}$?

Are there some known results for $\sum\limits_{p\text{ prime}} \dfrac{1}{p^2-1}$? I wasn't able to find anything about this sum in the internet or in my books!
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1answer
7 views

Solving a Extended Euclidean Algorithm related problem

Alex has some (say, n) marbles (small glass balls) and he has going to buy some boxes to store them. The boxes are of two types: Type 1: each box costs c1 Taka and can hold exactly n1 marbles Type ...
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2answers
22 views

Whether an equation has a solution

Will the following equation have a solution in $\mathbb Z$? $n_1^2+n_2^2+n_1n_2=3$ for $n_1\neq n_2$
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2answers
56 views

Determinant value of $2 \times 2$ matrices

Let $a,b,c,d$ be integers such that $\dfrac ac \in \mathbb Q^+$\ $\mathbb Z^+ $ and $\dfrac bd \in \mathbb Q^- $ \ $ \mathbb Z^-$ ; then how many solutions does $|ad-bc|=1$ have ?
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13 views

How can we solve the module function related equation?

Suppose that $\alpha,\beta=1,2,\cdots,n_1n_2$, and they satisfy the equation $$ \beta-\textbf{mod}(\beta,n_2)=\alpha-\textbf{mod}(\alpha,n_2) $$ where $\textbf{mod(,)}$ is the module function as usual ...
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0answers
33 views

Can Legendre's theorem really help solve this equation: $ax^2+by^2=cz^2$?

let $a,b,c,x,y$ be non-zero positive integers such that $$\gcd(x,y,z)=1$$ $$ \gcd(x,a)>1$$$$ \gcd(y,b)>1$$ $$ \gcd(z,c)>1 $$ If $a,b,c$ are square-free, find all the non-trivial integral ...
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2answers
29 views

For any integer a, if $6|(3−a)$, then $3| (a−2)$.

Prove: For any integer a, if $6|(3−a)$, then $3| (a−2)$. I've been trying to work this problem for a while, but missed a day of class and can't seem to work it out.
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24 views

How many zeros within a number

Let noughts(n) be the number of noughts needed to write n in base 10.If n is given how can I find out the value of noughts(n) . I myself have tried to compute noughts(n) by examining all the digits ...
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4answers
91 views

Solve this number theory problem

Why is a number written in decimal evenly divisible by 9 if and only if the sum of its digits is a multiple of 9?
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2answers
228 views

Proving a palindromic integer with an even number of digits is divisible by 11

I'm in an introductory course for discrete math so I'm a novice at English proofs. I'm not sure if my reasoning here is valid or if I'm using modular arithmetic correctly. Specifically the line I ...
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0answers
22 views

Remarks on a Previous Post

Recently I have been reading this post and I have noted something significant in the fake argument. As one can easily see that the basic idea behind the argument had been to show that the sequence ...
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3answers
50 views

Prove there do not exist natural numbers m and n such that $7m^2 = n^2$.

Prove there do not exist natural numbers $m$ and $n$ such that $7m^2 = n^2$. Proof: Using the Fundamental Theorem of Arithmetic, we can write $m=(p_1^{r_1 }\ldots p_n^{r_n})$ and $n=(q_1^{s_1 }\ldots ...
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0answers
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$n_p$ - the largest power of the prime $p$ which divides $n$

I was reading this article called "On A Theorem of Frobenius: Solutions to $x^n=1$ in Finite Groups" by I.M. Isaacs and G.R. Robinson (www.jstor.org/stable/2324902). In the third para of the first ...
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1answer
27 views

Numbers of the form $n^k-1$

I know that numbers of the form $2^k-1$ are called Mersenne numbers. But are there other special numbers which are one less than a power of an integer (for instance, does $3^k-1$ have some special ...
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1answer
50 views

Fermat's Little Theorem: group and multiplication modulo

$p$ is a prime number. $G$ is a group of integers $\{1,2,\dots,p-1\}$ under multiplication mod $p$. $d$ is a divisor of $(p-1)$ Is it possible to prove that the number of elements $a$ in $G$ such ...
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1answer
12 views

Gaussian sums values

I have the following problem: Denoting $S(q,a,\chi ) = \sum_{x=1}^q \chi (x) e(ax/q)$, where $\chi $ is an arbitrary character modulo $q$, I have to prove $$\sum_{a=1}^q \vert S(q,a,\chi ) \vert ^2 = ...
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1answer
24 views

How many ordered bases can be found for $\mathbb{Z}_p^n$ over filed $\mathbb{Z}_p$?

Take $\mathbb{Z}_p^n$ as a linear space over $\mathbb{Z}_p$. Now you can imagine multy bases for this space. (please leave a comment or have an edit if question is not clear enough.)
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1answer
14 views

reducing the modulus of a Dirichlet character

Let $\chi$ be a Dirichlet character modulo $N$. Let $M$ be a positive divisor of $N$ such that $$\text{radical}(N)=\text{radical}(M).$$ Is $\chi$ be a character modulo $M$? Best regards.
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1answer
25 views

Modular arithmetic - is this a “legal” substitution?

I know that $$a \equiv b ~(\text{mod}~3)$$ and $$c \cdot a \equiv 1 ~(\text{mod}~3)$$ Can I substitute $a$ with $b$? I mean: $$c\cdot b \equiv 1 ~(\text{mod}~3)$$
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1answer
20 views

Dirichlet character modulo p

How can I prove that if $\chi$ is a non-principal character modulo $p$ prime, then $\chi (-1) = \overline{\chi} (-1)= \pm 1$ and $\sum_{x=1}^p \chi (x) e^{2\pi i x}=0$? For the first question, I just ...
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0answers
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no quotient of $\mathcal{O}_{\mathbb{C}_K}$ modulo a proper ideal containing $p$ is a perfect $\mathbb{F}_p$-algebra?

I am reading the Notes on $p$-adic Hodge theory of O. Brinon & B. Conrad . Can someone explains the following things to me? «... no quotient of $\mathcal{O}_{\mathbb{C}_K}$ modulo a proper ideal ...
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68 views

Cardinality of a set of polynomials where the sum of the cubes of the roots is zero

Let $C\subseteq \mathbb Z\times \mathbb Z$ be the set of integer pairs $(a,b)$ for which the 3 complex roots $r_1,r_2,r_3$ of the polynomial $p(x)=x^3-2x^2+ax+b$ satisfy $r_1^3+r_2^3+r_3^3=0$ .Then ...
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1answer
36 views

$t^{\frac{1}{t-1}}$ and $t^{\frac{t}{t-1}}$ need to be integers

$t^{\frac{1}{t-1}}$ and $t^{\frac{t}{t-1}}$ need to be integers. Is this only for $t = 2$ and $t = \frac{1}{2}$ true? $t$ can be any positive real number Could anyone give me a hint how to prove it, ...
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2answers
64 views

Prove that there exist a prime having last $65050$ digits the largest known prime

The largest known prime is of the $65050$ digits. Prove that there exist another prime which ends in the same $65050$ digits of largest known prime.
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1answer
18 views

if $p=(a+ib)(c+id)$ and $p^2 = a^2 + b^2$ then $p\mid a$ & $p\mid b$

We're working on Gauss integers... p is an odd prime such that $p \not\equiv 1 \pmod 4$. We want to prove that if there is $(a,b,c,d) \in \mathbb{Z}^4$ such that $$p = (a+ib)(c+id) \text{ ...
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0answers
39 views

Pell's Equation and the Pigeon Hole Principle

David Speyer gave a beautiful application of the pigeon hole principle here to show that Pell's equation $$x^2-Dy^2=1$$ has infinitely many integral solutions. I was wondering if anybody knows the ...
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9 views

Twisted logarithm power series

I recently encountered a power series similar to the one of the $\log(1-x)$ of the form $$ F(x)= \sum_{n=1}^\infty \frac{\psi(n)x^n}{n}, $$ where $\psi$ is some Dirichlet character. Has anyone here ...
2
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1answer
35 views

Average order of Eulers totient function squared

I was wondering if one has a nice asymptotic formula for the sum $$\sum_{n\le x} \phi(n)^2$$ and if so, how does one calculate it. I know that one has $\sum_{n\le x} \phi(n) = ...
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1answer
33 views

Is $r_2$ a uniformly at random value in $Z_n$, where $r_2=r_1 . m$

Let $m$ be an arbitrary value in $Z_n$, where n is RSA modulo (n=p.q, where p and q are large primes). Then have: $r_2=r_1 . m$, where $r_1$ is a value chosen uniformly at random : $r_1\in Z^*_n$. ...
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0answers
28 views

Concise Number Theory Book [duplicate]

I'm having problems relating different formulas etc with each other in number theory. So I need recommendations for a really good book on introduction to number theory which is precise and has plenty ...