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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Number analysis

I was preparing for SAT, and saw this question (without answers) and was stumped. If 2015 = 101a + 10b, what is a + b? I broke 2015 into primes, one of which is 5, and if you break 10 into primes ...
3
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1answer
30 views

Proving that the multiplicative group mod p (p is prime) is cyclic

Let $\mathbb{Z}_p^*$ be the group of integers $\{1,2,3,\dots,p-1\}$ under multiplication mod $p$, where $p$ is a prime. Given the following two facts: If $d$ divides $p-1$, then there are exactly ...
4
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0answers
55 views

Representing of natural number

Let $n$ be a given natural number with $n>1$, please prove that each natural $x>=2n$ can be represented as $$x=a_1^n+a_2^n+\cdots+a_n^n-(b_1^n+\cdots+b_{n-1}^n)$$ where ...
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35 views

If $a-b,a+b,b+c,b-c,a-c,a+c$ are square number,Find minimum of the $a+b+c$

let $a,b,c$ be postive integer numbers, such $a>b>c$, and $$a-b,a+b,b+c,b-c,a+c,a-c$$ be square number. Find the $$(a+b+c)_{min}$$ My idea: let $$a-b=p^2,a+b=q^2\Longrightarrow ...
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1answer
141 views

$2 \times 3 = 5+1$ and $2+3 = 5 \times 1$. When else can we switch the operators like this?

I noticed the following: $$2 \times 3 = 5+1$$ If you switch the operators, it is still true: $$2+3 = 5 \times 1$$ There is another obvious/trivial example where you can swap the operators: $$2\times 2 ...
2
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1answer
45 views

Prime Factorization and Number Theory

Prime factorization of $n$ is $$n = p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$$ Let $f(n) = p_1^{e_1}p_2^{e_2}p_3^{e_3}\cdots p_k^{e_k}$ where $e_k=a_k$ if $p_k|a_k$, else $e_k=a_k-1$ I want to ...
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5answers
46 views

Prove that $gcd(a, b) = gcd(b, a-b)$

I can understand the concept that $\gcd(a, b) = \gcd(b, r)$, where $a = bq + r$, which is grounded from the fact that $\gcd(a, b) = \gcd(b, a-b)$, but I have no intuition for the latter.
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0answers
11 views

a question on upper bound for Bessel function $K_{2it}(x)$

Can we have $$K_{2it}(x)\sinh(t)\ll_{x} 1$$ for $1<x< (1+|t|)^3,$ where $K_{2it}(x)$ deotes the ordinary K-bessel function and $t>1$. This is true when $x\ge (1+|t|)^3$ from some references. ...
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2answers
26 views

For part (ii), how can I prove that using Fermat's Theorem?

In this question, I think I know how to do part (i), $p^q+p^q-p-q=p(p^{q-1}-1)+q(q^{p-1}-1) $and then using Fermat's Theorem. In part (ii), I try to do $p^{q^3+q}= p^{q(q^2+1)}$, but I do not know ...
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1answer
60 views

Consequences of the negation of the Riemann hypothesis

There are many sources documenting the consequences of the Riemann hypothesis, but I can't find one discussing the consequences of its negation, particularly concerning the prime distribution.
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1answer
11 views

Concrete and elementary applications of modular forms to elliptic curves

What are some useful facts/algorithms for elliptic curves that can be obtained (proved completely) using the theory of modular forms without heavy machinery? It's often been asked what elementary ...
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25 views

about numer thorey [on hold]

For each positive integer m define Zm = {0, 1, 2, . . . , m − 1}, the set of all residues modulo m, and define C(m) = { k ∈ Zm | 0 not equal to k ≡ a^3 (mod m) for some a ∈ Z } the set of mod m ...
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0answers
12 views

Syndeticity and A.P.-richness of certain sets

Let $A \subset \mathbb{N}: \sum_{a \in A} (\frac{1}{a}) = \infty$; denote $\{ \alpha_1 @ \alpha_2: \alpha_1, \alpha_2 \in A \} = A @ A$, where "$@$" is any appropriate binary operator. (Note: $A$ is ...
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7 views

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
15 views

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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1answer
34 views

how to predict the sum of digits of the number $A(n)$ for a large natural $n$ without calculation, when $A(n)=a(n^2+n)+b$?

look $A(n)=9n^2+9n-1$ , 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 $k$ ...
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2answers
17 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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18 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
14 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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20 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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1answer
43 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
30 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
44 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
70 views

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

Find all integers $x$ and $y$ such that $x^5-27y^3=2x$.
3
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1answer
39 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
83 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
43 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
50 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
30 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
33 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
61 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
15 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
50 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
8 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
23 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
57 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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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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2answers
60 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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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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25 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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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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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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51 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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19 views

$n_p$ - 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
64 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
13 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
15 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.