Questions tagged [number-theory]

Questions on advanced topics - beyond those in typical introductory courses: higher degree algebraic number and function fields, Diophantine equations, geometry of numbers / lattices, quadratic forms, discontinuous groups and and automorphic forms, Diophantine approximation, transcendental numbers, elliptic curves and arithmetic algebraic geometry, exponential and character sums, Zeta and L-functions, multiplicative and additive number theory, etc.

Filter by
Sorted by
Tagged with
2 votes
0 answers
15 views

Number of Solutions of the Hyperbola Equations over Finite Fields

I have a problem with proving the number of points of the hyperbola equation $H_a: x^2 + y^2 = a$ (for every a > 0 in finite field $F_p$) in the finite fields. I have to prove that the number of ...
user avatar
  • 21
-3 votes
1 answer
25 views

Does there exists positive integers $m,n$ with $m\leq n$ and ${}_{n}m^k$ contains all of the $n$ digits for all sufficiently large integer $k$'s??

Let ${}_{n}x$ denote the base-$n$ ($n\geq 2$) representation of the integer $x$. Does there exists positive integers $m,n$ satisfying the following conditions $m\leq n$ $\exists k_0\gt 0$ such that ...
user avatar
  • 1,651
0 votes
1 answer
31 views

Silverman Proposition 2.5 computation

In the proof of Proposition 2.5 in Silverman's Arithmetic of Elliptic Curves, the author defines a map $$E_{ns} \to \overline{K}^*, \quad [X,Y,Z] \mapsto 1 + \frac{AX}{Y},$$ where $E_{ns}$ is the ...
user avatar
0 votes
0 answers
21 views

Tate-Shafarevich group and number of points $\bmod p$

Is there any known relationships or conjectures between the size of the Tate-Shafarevich group and the number of points of the associated elliptic curves over $\mathbb{F}_p$. Thanks!
user avatar
0 votes
0 answers
26 views

How to simplify the expression for Dirichlet inverse of $\varphi$ further?

$\varphi$ : Euler totient function $\mu$: M$\ddot{o}$bius function $I =\chi_{\{1\}}$ $N\in\Bbb{K}^{\Bbb{N}}$ : $N(n)=n$ $u $ : unit function i.e $u(\Bbb{N}) =\{1\}$ We know $\varphi(n) =\sum_{d|n}\mu(...
user avatar
  • 4,246
2 votes
1 answer
112 views

What is the general number of solutions of the equation $y^2=x^3-x\pmod{p}$?

What is the general number of solutions of the equation $y^2=x^3-x\pmod{p}$, where $p$ is a prime number and $p>3$?. Calculations suggest that the number of solutions to this equation is $p$ if $p\...
user avatar
5 votes
1 answer
93 views

Is integer part of $e^n$ infinitely often even and odd?

Let $a_n:=\lfloor e^n\rfloor$. Are $\{n\mid 2|a_n\},\,\{n\mid 2|a_n-1\}$ both infinite sets? More generally, for any irrational number $\alpha>1$, is each set of the form $\{n\mid p|\lfloor\alpha^n\...
user avatar
0 votes
1 answer
19 views

Tate gamma factor as a principal value integral

Let $F$ be a local field, $\chi$ a multiplicative character of $F^{\ast}$, and $\psi$ an additive character of $F$. The gamma factor $\gamma(s,\chi,\psi)$ is defined by means of the local functional ...
user avatar
  • 30.5k
-5 votes
0 answers
42 views

Math Questions Basic level [closed]

Find number of integers n such that n+3 divides n^3-3 ?
user avatar
3 votes
1 answer
59 views

A function asymptotical equivalent with the prime counting function?

Let $p_n$ be the $n$-th prime number and $Q_a(N)$ be the number of primes of the form $p_n^2+a$ where $1\leq n\leq N$ and $a$ is positive and even. For some $a$ like $26,56$ it seems that no solutions ...
user avatar
  • 13.3k
1 vote
1 answer
39 views

Proof regarding principal factors of the discriminant in $\mathbb{Q}(\sqrt{d})$

So I understand there are (up to $\pm$) exactly two primitive (no rational integer factors) elements $\alpha_1 ,\alpha_2 \in \mathcal{O}_K$ such that the fundamental unit $\varepsilon$ of $K=\mathbb{Q}...
user avatar
  • 11
2 votes
1 answer
54 views

For every prime p there is a sum of squares congruent to -1 mod p [duplicate]

For every prime $p$, there exists $a,b \in \mathbb{Z}$ such that $p\mid a^2+b^2+1$ For context, this question shows up as a statement on a hint to showing that every positive integer is a sum of 4 ...
user avatar
-3 votes
0 answers
30 views

Mobius inversion formula proof [closed]

This is a part of the proof for mobius inversion formula I don't understand how do we get the second and third line. Please explain \begin{aligned} \sum_{d|n} \mu(d) F(n/d) &=& \sum_{d|n}\mu(d)...
user avatar
0 votes
1 answer
49 views

Show: $\text{lcm}(a,a+p)=\text{lcm}(b,b+p), p \;\text{prime}\implies a=b$

(Romania Mathematical Olympiad). Let $a,b$ be positive integers such that exists a prime $p$ with the property $lcm(a,a+p)=lcm(b,b+p)$. Prove that $a=b$. What I could do: WLOG $p|a, p \nmid b \...
user avatar
0 votes
0 answers
38 views

Books about operator algebra and number theory

Does anyone know books that covers both operator algebras and number theory. Actually, a number theory books that has operator algebraic approaches.
user avatar
3 votes
1 answer
89 views

n divides $m+1, m^m+1, m^{m^m}+1,...$

Prove that for each positive integer n, there is a positive integer m such that each term of the infinite sequence $m+1, m^m+1, m^{m^m}+1,...$ is divisible by n. The only thing I could work out was ...
user avatar
0 votes
1 answer
23 views

Integers starting at 1 and ending at 5 that are divisible by 9.

I want to find the integers starting at 1 and ending at 5 that are divisible by 9. An integer is a multiple of 9 if: $$n = a_k a_{k-1} \cdots a_1 a_0 \Rightarrow a_k+a_{k-1}+\cdots a_0 \equiv 0 \pmod{...
user avatar
0 votes
0 answers
23 views

Is this part of the Banach density function $BD$ subadditive?

Let $A$ be a set and consider the function $$f(n) = \max_{x \in \mathbb{Z}} |A \cap [x+1,x+n]|,$$ where $|\cdot|$ denotes cardinality and $n \in \mathbb{N}$. I am reading a paper "An elementary ...
user avatar
0 votes
0 answers
64 views

$n=s^2-t^2$ how many values can $s$ take?

Let's say we have $n=pq$ with $p,q$ prime. We can write $n=s^2-t^2$ for some whole numbers $s,t$. Now prove that if $q<p\leq (1+\epsilon)\sqrt{n}$ then $s$ has at most $\frac{\epsilon^2}{2}\sqrt{n}$...
user avatar
  • 1,406
1 vote
1 answer
173 views

Three consecutive powerful integers do not exist

I (who is not a professional mathematician), ended up in the following on which I would like to have your comment, because this is overly simple solution. I know this certainly should not be easy and ...
user avatar
  • 21
1 vote
0 answers
20 views

The Ramification of the Torsion Field of the Frey Curve

I seek to correct my proof of the following well-known fact about the Frey curve, and to ask a few questions. References are acceptable answers except for the last bold request; I would like to ...
user avatar
  • 3,567
2 votes
1 answer
32 views

Evaluate/asymptotic of the sum $\sum_{a = 1}^{L \left({N}\right)} \left({- 1}\right)^{\left\lfloor{N/\left({2\, a + 1}\right)}\right\rfloor}$

I am trying to evaluate the sum and its asymptotic limit as $N \rightarrow \infty$ of $$\sum_{a = 1}^{L \left({N}\right)} \left({- 1}\right)^{\left\lfloor{N/\left({2\, a + 1}\right)}\right\rfloor}$$ ...
user avatar
1 vote
0 answers
25 views

Global Langlands for tori?

Where can I find a clear statement (perhaps with some proof details and original references) for the Langlands functoriality for tori, or alternatively, where was this originally proved by Langlands?
user avatar
  • 777
16 votes
2 answers
483 views

Collatz conjecture but with $n^2-1$ instead of $3n+1.$ Does the sequence starting with $13$ go to infinity?

Let's consider the following variant of Collatz $(3n+1) : $ If $n$ is odd then $n \to n^2-1.$ $1\to 0.$ $3\to 8\to 1\to 0.$ $5\to 24\to 3\to 0.$ $7\to 48\to 3\to 0.$ $9\to 80\to 5\to 0.$ $11\to 120\to ...
user avatar
0 votes
0 answers
150 views

(Novel?) sieve of co-primes that contains all primes. [closed]

Question Per some individuals request I have to phrase this as a focused question for this Q&A site. So the main question is: Is the following sieve and conclusion novel? All prime numbers are ...
user avatar
  • 133
0 votes
0 answers
38 views

Meaning of Big O notation with Index $[O_{\epsilon}(n^{\epsilon})]$

I found this notation in a book and can't figure what it means. I haven't found any other examples. This is $$O_{\epsilon}(n^{\epsilon})$$ One can see it in context at page $186$ of book. Anyone know ...
user avatar
0 votes
0 answers
29 views

Harmonic Distribution of prime numbers [closed]

I developed a sieve that depicts the distribution of prime numbers as contained in harmonic (repetitive) patterns. Published it here What would be the process to know if I’m rightfully thinking this ...
user avatar
  • 133
2 votes
2 answers
107 views

Are $\frac{p^2+1}{2}$ and $\frac{p^{5n}(p^5-1)}{2}$ are coprime to each other, $n \in \mathbb{N}$?

Let $p$ be a prime integer greater than $2$. Then I want to prove the followings: $(1)$ $\frac{p^2+1}{2}$ and $\frac{p^5-1}{2}$ are coprime to each other. $(2)$ $\frac{p^2+1}{2}$ and $\frac{p^{5n}(p^5-...
user avatar
  • 9,587
1 vote
1 answer
76 views

Find solution using infinite descent.

Can someone help with a task? Need to find a solution other than $(0,0,0) $ with infinite descent. $x,y,z\in\mathbb{Z}$. Any help would be appreciated. The equation is $x^2-3y^2=2z^2$. I tried to ...
user avatar
  • 23
6 votes
3 answers
295 views

Why does this cycle of 44 show up in the Collatz Conjecture?

Consider this function: $$f\left(x\right)=\frac{x-b^{\left(\operatorname{floor}\left(\log_{b}x\right)\right)}}{b^{\left(\operatorname{floor}\left(\log_{b}x\right)\ +\ 1\right)}-b^{\left(\operatorname{...
user avatar
  • 99
8 votes
0 answers
123 views

Positive integers satisfying $a^b = cd$ and $b^a = c+d$

Yesterday, at 23:18, I thought it was a remarkable moment of the day. The digits on the watch were providing a quadruplet of positive integers that satisfy the following system of equations: $$\begin{...
user avatar
0 votes
0 answers
21 views

2 questions in the proof of Brun Titchmarch Inequality

This question is from lecture 13 of the notes of Sieve Theory here:http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html I have 2 questions in the proof of lemma 2.2 on page 3: Question 1 : I am ...
user avatar
  • 1,344
0 votes
0 answers
45 views

find all irreducible polynomials of degree 2 and 3 over Z5

I would like to find all irreducible polynomials of degree 2 and 3 with coefficients in Z5. I know that the polynomial (x^5)^n - x equals with the product of all monic irreducible polynomials of ...
user avatar
-2 votes
1 answer
38 views

Questions in theorem related to primes with fixed modulus

This question is from notes on sieve theory here:http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html. I have questions in page 4 of lecture 12(http://www.math.tau.ac.il/~rudnick/courses/...
user avatar
  • 1,344
-1 votes
1 answer
101 views

Find the invers of $4 \in \mathbb{Z}_5$ (The 5-adic integers)

I am trying to solve this question, however I don´t seem to have the correct expression of the inverse to solve the remaining part: QUESTION: Find the inverse of 4 in $\mathbb{Z}5$. Use your answer to ...
user avatar
-3 votes
0 answers
37 views

A question in a theorem related to bounding primes values of $n^2 +1$ [closed]

This question is in the proof of theorem 2.1 in Lecture 12 of the notes of Sieve Theory here:http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html I have checked in the notes of earlier lecture ...
user avatar
  • 1,344
0 votes
0 answers
95 views

Attempting to restate the question of whether the collatz conjecture has a nontrivial cycle as a combinatorics problem

It occurs to me that the question about whether non-trivial cycles exist for the collatz conjecture can be restated as these two questions (details on how this relates to the collatz conjecture can be ...
user avatar
3 votes
1 answer
94 views

Find a polynomial of the form $F(x,y,z)$ of degree $3$ such that $F(a,b,c) = 0 \pmod{5}$ iff $a,b,c= 0 \pmod{5}$

I am trying to solve this question to study for my Number Theory final exam QUESTION: Find a polynomial of the form $F(x,y,z)$ of degree 3 such that $F(a,b,c) \equiv 0 \pmod{5}$ iff $a,b,c \equiv 0\...
user avatar
-2 votes
0 answers
68 views
+150

Questions about Hooleys Approach in Artin's primitive root Conjecture

I know that I have asked 5 questions but they are all part of same proof. To each answer that answer 3 or more questions I will grant a bounty of 100 points and if someone answers all 5 questions I ...
user avatar
  • 1,344
0 votes
2 answers
42 views

What are the fixed points of the arithmetic derivative over the non-negative integers?

I just watched When the derivative of a number is not zero -- The arithmetic derivative by Michael Penn. I like to explore things visually and computationally, so I found this recursive implementation ...
user avatar
  • 1,316
0 votes
0 answers
25 views

Perfect squares can't be primitive roots [duplicate]

This question was asked in my assignment on number theory and I am struck on this. Question: Prove that perfect squares can't be primitive roots if p>2. Attempt: let a is a perfect square and on ...
user avatar
  • 69
-2 votes
0 answers
61 views

Primitive root and prime $p'$ such that $4p' +1$ is also a prime [closed]

The following question is from my assignment in number theory and I am not able to make any progress on this. I have been following Elementary number theory by David Burton. If $p$ is a prime of the ...
user avatar
  • 69
1 vote
1 answer
44 views

2 questions in the theory of Counting Perfect Squares

I have been reading sieve theory from notes of zeev rudnick here :http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html and in page 2 of lecture 16(http://www.math.tau.ac.il/~rudnick/courses/...
user avatar
  • 1,344
1 vote
1 answer
50 views

$M_k(\Gamma_1(N)) = \bigoplus_{\chi \mod N} M_k(N, \chi)$

I am stuck with the following statement in the study of modular forms: $$ M_k(\Gamma_1(N)) = \bigoplus_{\chi \mod N} M_k(N, \chi), $$ where $\Gamma_1(N) := \left\{\begin{pmatrix}a & b\\ c & d\...
user avatar
  • 366
0 votes
0 answers
24 views

Digits tower power iterate

Stack the digits of a natural number into a power tower, iterate until only one digit remains. Does this iteration always terminate for any positive integer? Additionally specify $0^n = 0$, even when $...
user avatar
  • 843
0 votes
0 answers
37 views

Show that an analytic function is always negative

I want to show the following function is negative for $z\in [0,1)$: $$f(z) = -1 + z^2(z-1) + 2\sum_{k=0}^\infty (-1)^k z^{(2k+1)^2+1}. $$ By Tauberian theorem, I know that $\lim_{z\to 1^-}f(z)=0$. We ...
user avatar
2 votes
0 answers
72 views

Is there a closed-form expression for $f(x)=\sum\limits_{k=1}^\infty (\text{Ei}(i k x)+\text{Ei}(-i k x))$ assuming $x\in\mathbb{R}$?

Question: Is there a closed-form expression for $$f(x)=\sum\limits_{k=1}^\infty (\text{Ei}(i k x)+\text{Ei}(-i k x))\,,\quad x\in\mathbb{R}\tag{1}$$ where $\text{Ei}(z)$ is the exponential integral ...
user avatar
  • 4,538
5 votes
1 answer
65 views

Is $(1+c^2)^n-\lfloor(1+c^2)^{n/2}\rfloor^2<(1+c^2)^{(n+1)/2}$ true for all integers $c>1$, when $n$ is an odd integer?

Let $n$ be an odd integer. Is $$(1+c^2)^n-\lfloor(1+c^2)^{n/2}\rfloor^2<(1+c^2)^{(n+1)/2}$$ true for all integers $c>1$? Notes: $c=1$ has a counterexample $2^{31}-\lfloor2^{31/2}\rfloor^2>2^{...
user avatar
  • 25.2k
-1 votes
0 answers
17 views

How to prove this deduction in the Analytic Large Sieve using Beurling - Selberg function

I have been studying sieve theory from the following notes : http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html and I am struck on the following deduction in the proof of theorem 3.1 ( page 6) ...
user avatar
  • 1,344
0 votes
1 answer
24 views

Some questions in the proof of Analytic Large Sieve

I am learning about the analytic large sieve from the lecture notes here:http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html . I have some question in lecture 15:http://www.math.tau.ac.il/~...
user avatar
  • 1,344

1
2 3 4 5
767