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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1
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1answer
25 views

is Legendre Symbol related with inverse?

I want to show that If $p$ is an odd prime, then $\sum_{ j = 1 }^{ p-2 } \left(\frac{j^2 + j}p \right) = -1$ (where $\left(\frac\cdot\cdot\right)$ is Legendre Symbol). My teacher said I should ...
5
votes
2answers
59 views

Prove or disprove that $p_n > e^{p_n - p_{n-1}}$ for large enough $n$.

Let $p_n$ denote the $n$-th prime. Prove or disprove that for large enough $n$ we have $$p_n > e^{p_n - p_{n-1}}.$$ The inequality trivially holds for all the twin primes larger than $7$. With ...
3
votes
1answer
28 views

Modular arithmetic with huge modulus?

When the dividend is some huge power but the modulus is not so big, I can use modular exponentiation. But how can I compute the residue when the modulus is, for example, $2^{107} - 1$, a Mersenne ...
3
votes
0answers
71 views

Group generated by two polynomials

The $Homeo(\mathbb{R})$ be the group of all the homeomorphisms on $\mathbb{R}$, with the group operation of composition. Let $f(x)=x^3+\alpha$ and $g(x)=x^3+\beta$ be two elements of ...
2
votes
0answers
47 views

Finding newforms with Sage

I am new to Sage and modular forms. I have some conceptual questions. When I write sage: S = CuspForms(Gamma0(55),2,prec=14) sage: S.new_subspace().basis() ...
0
votes
0answers
26 views

ALGORITHM Multiplication of Integers from “discrete math and its applications 7th edition ” book

Please can you help me to understand the "italic text" How many additions of bits and shifts of bits are used to multiply a and b using Algorithm 3"see the the attached photo"? Solution: ...
6
votes
1answer
165 views

How can I know if $2^{2^{2^{2^{2}}}}+1=?$ is prime?

I could calculate the following prime numbers $$2+1=3$$ $$2^{2}+1=5$$ $$2^{2^{2}}+1=17$$ $$2^{2^{2^{2}}}+1=65537$$ Are the following numbers prime??? $$2^{2^{2^{2^{2}}}}+1=?$$ ...
2
votes
1answer
44 views

Does it have convergent subsequence in that form? [duplicate]

Let $P_i$ be sequence of prime numbers i.e $P_1=2,P_2=3,P_3=5 ...$ Euler has proved that the sum $$\sum_{i=1}^{\infty}\dfrac{1}{P_i}$$ is divergent. Set $a_i=P_{P_i}$ then $a_1=P_2=3$ , ...
9
votes
2answers
107 views

Proving $(\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}})(\sqrt{x-\sqrt{x-\sqrt{x-\sqrt{x+\cdots}}}})=x$

How can I prove this equality? $$\left(\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}}\right)\left(\sqrt{x-\sqrt{x-\sqrt{x-\sqrt{x+\cdots}}}}\right)=x$$ if $$x>1$$
0
votes
1answer
23 views

Characterization of integral quadratic forms representing the same numbers? [duplicate]

Is there a simple characterization of integral quadratic forms that represent the same numbers? I know that if two quadratic forms are in the same $GL_n(\mathbb{Z})$-orbit then they represent the ...
2
votes
3answers
40 views

How does Euler's totent function use p is not equal to q?

I am writing a paper on RSA and introducing Euler's totent function. I wrote $\phi(n) = \phi(pq)= \phi(p) \times \phi(q) = (p - 1) \times (q -1)$ and my teacher said it is FALSE if p = q. He then ...
1
vote
1answer
25 views

Sum of all elements of a Set

Let's say I want to determine the number of natural numbers for an $x \in N$ this particular way: $$f(x) = \sum_{i=1}^x a\in\lbrace 1 : x\space\mathbf {mod}\space i=0\rbrace$$Is this the correct way ...
0
votes
0answers
24 views

Order of generator

If i choose $p = 2q + 1$ and i have a generator/primitive element $g \in \mathbb{F}_p^*$, the order of $ord\left( {{g^q}} \right) = ord\left( {{g^{\frac{{p - 1}}{2}}}} \right) = 2$. Now the ...
0
votes
3answers
31 views

How can I compute the value of the following Legendre symbols?

I have read more examples how can I compute the value of Legendre symbols, but I can't find the right way with these examples. The prime factorisation have to help, but I can't apply it in these ...
0
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1answer
13 views

How can I proof the following properties of the Legendre-symbol?

How can I prove these properties of the Legendre symbol?
-6
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2answers
122 views

Prove that this recurrence relation algorithm generates all positive rational numbers, and does so without repetition and in reduced form [closed]

For $n\ge 1$, generate a sequence $\{a_n\}$ such that for any even $n = 2k$: $$ a_n = a_k$$ And for any odd $n=2k+1$: $$ a_n = a_k + a_{k+1}$$ With initial conditions $a_1 = a_2 = 1$ Now, generate a ...
0
votes
2answers
11 views

Proving a property about quadratic residues

I am trying to determine which primes satisfy the equation $x^2+x\equiv y \pmod{p}$ for a given $y$. For example, the $y$ which have solutions for $p = 11$ are the set $\{0,1,2,6,8,9\}$. However, ...
10
votes
3answers
114 views

How can I prove $\sqrt{(111…)+(55…)^2}=5…6$

The formula in my question can be illustrated as follow: $$\sqrt{11+5^2}=6$$ $$\sqrt{111+55^2}=56$$ $$\sqrt{1111+555^2}=556$$ $$\sqrt{11111+5555^2}=5556$$ and so on How can I prove the general ...
8
votes
3answers
116 views

$x^n + y^n = c$ has finitely many integral solutions?

Assume $n > 1$ and $n$ is odd because it's easy if $n$ is even. Please help prove this. $x^n + y^n = c$ has finitely many integral solutions if $c \neq 0$? Thank you all for replying. I think ...
0
votes
1answer
29 views

The ring $\Bbb Z_p[\sqrt{d}]$ and its invertible elements

Let $\Bbb Z_p[\sqrt{d}] = \{a + b\sqrt{d} : a,b \in \Bbb Z_p\}$, where $p$ is a prime and $d$ and $p$ are coprime.. I wish to show that if $d$ is not a quadratic residue modulo $p$, then $|\Bbb ...
-4
votes
1answer
56 views

How many integers from $1$ to $1000$ are divisible by none of $3 , 5$ and $7$ [closed]

answer me please How many integers from $1$ to $1000$ are divisible by none of $3 , 5$ and $7$
0
votes
3answers
48 views

How to solve this [duplicate]

A room contains 1001 bulbs each numbered from 1 to 1001. the bulbs are controlled by a toggle switch and all the bulbs are initially turned off. There are 1001 number theory experts each wearing a ...
1
vote
1answer
103 views

Does $\pi$ contain any zeroes?

Let's say we have two functions, $f$ and $g$. $f:\mathbb{R}\mapsto [0,1]$ where $0,1$ denote true, false respectively. $f(x)=1$ when $x$ contains any zeroes as a digit; $f(x)=0$ otherwise. Now let's ...
1
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0answers
21 views

A question on hilbert symbol in $Q_p$

Let $\alpha, \beta , \gamma$ are non-zero elements of $Q_p$, show that $$(\alpha\gamma,\beta\gamma)=(\alpha,\beta)(\gamma,-\alpha\beta)$$, where $(\alpha,\beta)=1 $ or $-1$ whether $X^2-\alpha ...
1
vote
1answer
18 views

John sold some books at $24 each, and used the money to buy some concert tickets…

John sold some books at 24 dollars each, and used the money to buy some concert tickets at $50 each. He had no money left over after buying the tickets. What is the least amount of money he could have ...
1
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0answers
27 views

(Eichler-Shimura Isomorphism) Proving c(f) is not a coboundary

I have seen a couple of questions related to the Eichler-Shimura Isomorphism, but almost all of them have to do with hodge theory (things I am unfamiliar with) and seem, to me, different/unrelated. ...
0
votes
0answers
11 views

Number of *distinct* dot products of an integer vector by elements of a hyper-rectangle

Imagine a vector $\boldsymbol{v}$ composed of integers, and the set $S$ of all integer vectors within a hyper-rectange, with one corner at the origin and other at $\boldsymbol{m}$. In other words: $S ...
0
votes
2answers
68 views

Is it possible to get the number of digits mathematically?

I was wondering if it was possible to, say, have function $f$ that would return the number of digits in any given positive integer. I tried using some sort of a summation, but that failed quite ...
0
votes
0answers
62 views

Parametric solutions to the Pell equation $x^2-dy^2=-4$?

I'm looking for identities for the fundamental solutions of, $$x^2-dy^2 = -4\tag{1}$$ The only one I know is, $$\begin{aligned} x \,&= m + 3 n + 2 m n^2 + n^3 + m n^4\\ d \,&= 4 + m^2 + 6 m ...
1
vote
0answers
29 views

Find out how the prime numbers 2, 3, 37 splits in K = Q(√37) [closed]

Find out how the prime numbers $2,3,37$ split in $K=\Bbb Q(\sqrt{37})$; i.e. find $r$ and thos $e_i,f_i$ for $1\leq i\leq r$. How do I solve this. Thanks!
2
votes
4answers
84 views

Easiest proof for $\sum_{d|n}\phi(d)=n$ [duplicate]

To prove $\sum_{d|n}\phi(d)=n$. What is the easiest proof for this to tell my first year undergraduate junior. I do not want any Mobius inversion etc only elementry proof. Tthanks!
2
votes
1answer
66 views

Improvements of Dusart's lower bound for $ e^\gamma \log x \prod_{p\le x} \frac{p-1}{p}$.

Let $\gamma$ be the Euler-Mascheroni constant. In this paper (Theorem 6.12) it is proved that for $x\ge 2793$, $$ e^\gamma \log x \prod_{p\le x} \frac{p-1}{p}> 1-\frac{1}{5 \left(\log ...
1
vote
1answer
32 views

proof that $n$ is prime or has prime factor $\leq \sqrt{n}$

apparently my attempt proof is wrong says the chat person will, so can you guys tell me how to fix please :) Show that any integer $n \gt 1$ is either a prime or has as a factor a prime $\leq ...
0
votes
1answer
30 views

How many possible sums of the digits of an n-digit number?

Suppose I have a seven-digit positive number (allowing leading zeroes): How might I go about finding the total number of possible sums of those seven digits? My first instinct was to say it's simply ...
0
votes
0answers
19 views

References for Hilbert symbols on $p$-adic fields

Can somebody give me some reference (Please not Serre, as it is too tough for me now) any reference for the basics and concepts on $p$-adic rings and fields and then gradually relating them to ...
0
votes
1answer
33 views

A simple question on p-adic fields

I have asked many question tonight on $p$-adic and I am still confused. So here is a very basic thing I want to know but nobody has cleared this doubt. It might be very silly, but please answer it. ...
4
votes
2answers
63 views

Given that there is at least one solution to $a^{2} + 2b^{2} = p^{11}q^{13}$, find how many integers solutions there are.

I cannot even begin this problem, given $ a, b \in \mathbb{Z}$ and $p,q$ odd prime numbers, given that there is a soltuion to the equation: $a^{2} + 2b^{2} = p^{11}q^{13}$, find how many solutions ...
4
votes
1answer
46 views

Which $p$-adic fields contain these numbers?

Question: Determine the $p$-adic fields which contain $$ a)\;\sqrt{-1} \qquad b)\;\sqrt{3} \qquad c)\;\sqrt{-7} \qquad d)\;\sqrt{17}$$ I have no idea on this as I am completely confused with ...
-3
votes
2answers
32 views

Number theory / division algorithm

A par number is a integer n such that 2|n , a odd number is a integer that is not par. We said that the par Numbers have a parity 0 and odd Numbers parity 1. Assume that $z$ belong to the integers ...
0
votes
1answer
33 views

Linear congruence solution confusion

Why does: $4x \equiv 2 \bmod 6 $ have the solutions: $x \equiv 2 \bmod 6 $ AND $x \equiv 5 \bmod 6 $ I understand why $x \equiv 2 \bmod 6 $ as: $4 \cdot 2 = 8$ which is $2 + (1 \cdot 6)$ :. ...
3
votes
1answer
44 views

Two doubts about squares in $\Bbb Z_p$

The statement says that for $p \neq 2$ an element $x=p^i u \in \mathbb Q_p^\times$ (with $i \in \mathbb Z$ and $u \in \mathbb Z_p^\times$) is a square if and only if $i$ is even and $u$ is a square in ...
3
votes
2answers
80 views

Why does taking completions make number fields simpler?

I'm currently taking a course on Local Fields, and the local-theoretic picture seems to be significantly simpler than that of number fields. For example, If $K$ is a finite extension of $\mathbb ...
2
votes
1answer
39 views

Begginer doubt in Ring of p-adic integers

I am studying $p$-adic Rings and let me explain my understanding and doubt here. As I understood, Let $p$ be a rational prime and $Z$ denotes ring of integers, then form cartesian product $$P=Z/pZ ...
2
votes
1answer
39 views

Special case of Kronecker–Weber theorem.

Let $K$ be a number field contained in $m^{th}$ cyclotomic field, that is $K \subset \Bbb{Q}(\omega)$ where $\omega$ is a primitive $m^{th}$ root of unity. Let $p^k$ be the exact power of a prime $p$ ...
0
votes
1answer
23 views

Nontrivial characters of $(\Bbb{Z}_m)^{\ast}$

I was reading the book of Marcus on Number field page 196. I could not understang the highlighted equality. It will be helpful if someone gives me a proof. Thanks in advance for the computation!
-1
votes
1answer
29 views

Prove that 5 is a quadratic residue of an odd prime $p$

Prove that 5 is a quadratic residue of an odd prime $p$ if $p \equiv \pm1( mod 10)$, and that 5 is a non residue if $p \equiv \pm3 (mod 10)$.
0
votes
0answers
14 views

bounds for, $|L_{\tau}(s)|$, a Dirichlet searies associated with Ramanujan tau function

The Dirichlet searies associated with Ramanujan tau function is defined as: \begin{equation} L_{\tau}(s)=\sum_{n=0}^{\infty}\frac{\tau(n)}{n^s}=\prod_{p \text{ ...
1
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0answers
33 views

Consistent hashing using modulo

Following my answer here: Suppose I have n servers, and I want to distribute files evenly between them (same number of files on each server). Initially n=2 and I use the following function to map a ...
10
votes
2answers
668 views

Is the hypotenuse of a triangle ever divisible by three (for primitive Pythagorean triples)?

Looking for a proof that for primitive Pythagorean triples, the hypotenuse is never divisible by three. Below are a list of all the primitive Pythagorean triples with a hypotenuses less than 300. ...
0
votes
0answers
25 views

Define f:Z/3Z→Z/3Z by f([a])=[2a+1]

Just finished proving this to be injective, and well-defined. How would you prove it to be surjective? I understand surjective means that every element in the codomain is being used, and thus is the ...