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.
38,327
questions
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 ...
- 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 ...
- 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 ...
- 476
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!
- 599
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(...
- 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\...
- 29
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\...
- 79
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 ...
- 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 ?
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 ...
- 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}...
- 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 ...
- 464
-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)...
- 13
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 \...
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.
- 367
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 ...
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:
$$n = a_k a_{k-1} \cdots a_1 a_0 \Rightarrow a_k+a_{k-1}+\cdots a_0 \equiv 0 \pmod{...
- 33
0
votes
0
answers
23
views
Is this part of the Banach density function $BD$ subadditive?
Let $A$ be a set and consider the function $$f(n) = \max_{x \in \mathbb{Z}} |A \cap [x+1,x+n]|,$$ where $|\cdot|$ denotes cardinality and $n \in \mathbb{N}$. I am reading a paper "An elementary ...
- 505
0
votes
0
answers
64
views
$n=s^2-t^2$ how many values can $s$ take?
Let's say we have $n=pq$ with $p,q$ prime. We can write $n=s^2-t^2$ for some whole numbers $s,t$.
Now prove that if $q<p\leq (1+\epsilon)\sqrt{n}$ then $s$ has at most $\frac{\epsilon^2}{2}\sqrt{n}$...
- 1,406
1
vote
1
answer
173
views
Three consecutive powerful integers do not exist
I (who is not a professional mathematician), ended up in the following on which I would like to have your comment, because this is overly simple solution. I know this certainly should not be easy and ...
- 21
1
vote
0
answers
20
views
The Ramification of the Torsion Field of the Frey Curve
I seek to correct my proof of the following well-known fact about the Frey curve, and to ask a few questions. References are acceptable answers except for the last bold request; I would like to ...
- 3,567
2
votes
1
answer
32
views
Evaluate/asymptotic of the sum $\sum_{a = 1}^{L \left({N}\right)} \left({- 1}\right)^{\left\lfloor{N/\left({2\, a + 1}\right)}\right\rfloor}$
I am trying to evaluate the sum and its asymptotic limit as $N \rightarrow \infty$ of
$$\sum_{a = 1}^{L \left({N}\right)} \left({- 1}\right)^{\left\lfloor{N/\left({2\, a + 1}\right)}\right\rfloor}$$
...
- 749
1
vote
0
answers
25
views
Global Langlands for tori?
Where can I find a clear statement (perhaps with some proof details and original references) for the Langlands functoriality for tori, or alternatively, where was this originally proved by Langlands?
- 777
16
votes
2
answers
483
views
Collatz conjecture but with $n^2-1$ instead of $3n+1.$ Does the sequence starting with $13$ go to infinity?
Let's consider the following variant of Collatz $(3n+1) : $
If $n$ is odd then $n \to n^2-1.$
$1\to 0.$
$3\to 8\to 1\to 0.$
$5\to 24\to 3\to 0.$
$7\to 48\to 3\to 0.$
$9\to 80\to 5\to 0.$
$11\to 120\to ...
- 12.1k
0
votes
0
answers
150
views
(Novel?) sieve of co-primes that contains all primes. [closed]
Question
Per some individuals request I have to phrase this as a focused question for this Q&A site. So the main question is:
Is the following sieve and conclusion novel?
All prime numbers are ...
- 133
0
votes
0
answers
38
views
Meaning of Big O notation with Index $[O_{\epsilon}(n^{\epsilon})]$
I found this notation in a book and can't figure what it means. I haven't found any other examples. This is $$O_{\epsilon}(n^{\epsilon})$$ One can see it in context at page $186$ of book. Anyone know ...
- 11
0
votes
0
answers
29
views
Harmonic Distribution of prime numbers [closed]
I developed a sieve that depicts the distribution of prime numbers as contained in harmonic (repetitive) patterns.
Published it here
What would be the process to know if I’m rightfully thinking this ...
- 133
2
votes
2
answers
107
views
Are $\frac{p^2+1}{2}$ and $\frac{p^{5n}(p^5-1)}{2}$ are coprime to each other, $n \in \mathbb{N}$?
Let $p$ be a prime integer greater than $2$.
Then I want to prove the followings:
$(1)$ $\frac{p^2+1}{2}$ and $\frac{p^5-1}{2}$ are coprime to each other.
$(2)$ $\frac{p^2+1}{2}$ and $\frac{p^{5n}(p^5-...
- 9,587
1
vote
1
answer
76
views
Find solution using infinite descent.
Can someone help with a task? Need to find a solution other than $(0,0,0) $ with infinite descent. $x,y,z\in\mathbb{Z}$. Any help would be appreciated.
The equation is $x^2-3y^2=2z^2$.
I tried to ...
- 23
6
votes
3
answers
295
views
Why does this cycle of 44 show up in the Collatz Conjecture?
Consider this function:
$$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{...
- 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{...
- 560
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 ...
- 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 ...
-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/...
- 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 ...
- 39
-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 ...
- 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 ...
- 9,769
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\...
- 39
-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 ...
- 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 ...
- 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 ...
- 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 ...
- 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/...
- 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),
$$
where $\Gamma_1(N) := \left\{\begin{pmatrix}a & b\\
c & d\...
- 366
0
votes
0
answers
24
views
Digits tower power iterate
Stack the digits of a natural number into a power tower, iterate until only one digit remains.
Does this iteration always terminate for any positive integer?
Additionally specify $0^n = 0$, even when $...
- 843
0
votes
0
answers
37
views
Show that an analytic function is always negative
I want to show the following function is negative for $z\in [0,1)$:
$$f(z) = -1 + z^2(z-1) + 2\sum_{k=0}^\infty (-1)^k z^{(2k+1)^2+1}. $$
By Tauberian theorem, I know that $\lim_{z\to 1^-}f(z)=0$. We ...
- 329
2
votes
0
answers
72
views
Is there a closed-form expression for $f(x)=\sum\limits_{k=1}^\infty (\text{Ei}(i k x)+\text{Ei}(-i k x))$ assuming $x\in\mathbb{R}$?
Question: Is there a closed-form expression for
$$f(x)=\sum\limits_{k=1}^\infty (\text{Ei}(i k x)+\text{Ei}(-i k x))\,,\quad x\in\mathbb{R}\tag{1}$$
where $\text{Ei}(z)$ is the exponential integral ...
- 4,538
5
votes
1
answer
65
views
Is $(1+c^2)^n-\lfloor(1+c^2)^{n/2}\rfloor^2<(1+c^2)^{(n+1)/2}$ true for all integers $c>1$, when $n$ is an odd integer?
Let $n$ be an odd integer. Is $$(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^{...
- 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) ...
- 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/~...
- 1,344