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.

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-1 votes
1 answer
109 views

Sequence whose convergence towards 1 would be equivalent to the collatz conjecture being true, I would like feedback

I think I found a sequence whose convergence towards 1 would be equivalent to the collatz conjecture being true, we only need to study odd numbers since from an even number we can have an odd number ...
2 votes
1 answer
1k views

What is the maximum difference between these two functions?

Supposed the function $f(n)$ is defined as follows: For values of $n$ less than or equal to $121$ $$f(n)= \begin{cases} 0 ,& \text{if } n < 121\\ 1 ,& \text{if } n = 121\\ \end{...
3 votes
2 answers
61 views

Kummer's Lemma and $1+\zeta$

In lecture we were told to think about the following: Kummer's Lemma: Let $p$ be an odd prime and let $\zeta := e^{2\pi i / p}$. Every unit of $\mathbb{Z}[\zeta]$ is of the form $r\zeta^g$, where $r$ ...
0 votes
0 answers
4 views

Proving a canonical height property

Let $H$ be the classical Weil height for a number field (of degree $n$ say), i.e. given $\alpha$ an element of this field and $f(x) = a_0x^n + \cdots + a_n \in \mathbb{Z}[X]$ its minimal polynomial, ...
0 votes
0 answers
15 views

Coefficients of some polynomials factorization

Let $n,b\in\mathbb{N}$ and $P_{n_b}$ is polynomial whose coefficients are digits of $n$ in base $b$. Let $Q_{1}(x) \cdot Q_{2}(x) \cdots Q_{m}(x)$ - polynomial factorization (over integers) of $P_{n_b}...
0 votes
1 answer
24 views

Parametrization of rational points on the hyperboloid $z^2=-5x^2+4y^2+9$

I am trying to find rational parametrization of the rational points on the hyperboloid $z^2=-5x^2+4y^2+9$. I found many answers about parametrization of simple hyperbolic curves, but I cannot find the ...
3 votes
1 answer
224 views

On why solutions to $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$ also come in pairs

Jacobi and Madden found infinitely many primitive solutions to, $$a^4+b^4+c^4+d^4 = (a+b+c+d)^4$$ using an elliptic curve. We will use a different approach that, like the method for, $$a^4+b^4+c^4 = d^...
0 votes
0 answers
40 views

Are there any known results for Dirichlet series for trigonometric functions?

I'm currently working on a problem that required that I come up with a Dirichlet series for cos(s), where s is any complex number. After trying a couple different things, I arrived at the following ...
0 votes
0 answers
33 views

Existence of strictly increasing sequences such that $a_n(a_n+1) \mid b_n^2 + 1$.

Show that there exists two strictly increasing sequences $(a_n)$ and $(b_n)$ such that $a_n(a_n + 1) \mid b_n^2 + 1$. I have no idea how to solve this problem. I've tried a few different approaches, ...
0 votes
2 answers
384 views

system of congruences has a solution if GCD = 1

I have the following problem. Let m, n be arbitrary non zero integers. Show that $x ≡ a \pmod m$ $x ≡ b\pmod n$ has a solution if $\gcd(m, n)$ divides both $a$ and $b$. Also interested in the ...
7 votes
3 answers
8k views

modulo version of the quadratic formula and Euler's criterion

Use the modulo version of the quadratic formula and Euler's criterion to decide if the following has a solution or not. $2x^2+5x+8 \equiv 0\pmod{37}$ I'm not sure how I would use what was being ...
2 votes
0 answers
33 views

Bound on a sum of divisor function

I am currently reading a math paper in which the author writes : $$\sum_{l\mid n}l\sigma_1(l)\ll n^2\sigma_{-1}(n)$$ Where the function $\sigma_k$ is defined by $\sigma_k(n)=\sum_{d\mid n}d^k$ and ...
1 vote
2 answers
522 views

Closed Formula Expression for Sum of Combinatorics

I have recently been interested in the problem of summing Combinatorials. I have been beating my brain for the past days to figure out how to find an explicit closed form of: $n \choose 0 $+$ n \...
1 vote
0 answers
43 views

How well can you approximate $\mathbb{Z}$ using product of first $t$ numbers?

Is there a bound on how well you can (additive) approximate any natural number as the product of , say, first $10$ numbers - $\{1,2,3,\ldots,10\}$? By product, I mean just a single product, no ...
4 votes
1 answer
88 views

How many positive integers up to 100 can be written as a sum of 4 or more consecutive integers?

I was trying to solve the question: how many positive integers $n \leq 100$ can be written as the sum of 4 or more consecutive integers. Here was my approach: Let $m$ represent the number of ...
0 votes
0 answers
25 views

Partial summation and proving convergence

Let $a_n\in\mathbb{C}$ be a sequence such that there is a positive constant $\alpha$ with $\sum_{n\le x}a_n=O(x^{\alpha})$. Prove that for $\textrm{Re}(s)>\alpha$ we have that \begin{equation}\sum_{...
1 vote
1 answer
60 views

Show that a permutation equation has 5 solutions

How can I show that this permutation equation has 5 solutions: $\pi^{2013}$ = (1 9) (2 8) (3 7) (4 6) (5) Since the cycle structure is [2, 2, 2, 2, 1] then the only possible cyclic structure for the ...
0 votes
1 answer
48 views

Tamely extension of the maximal unramified extension of a local field

This construction comes from the chapter VII of "the local Langlands conjecture for $GL(2)$": Let $F$ be a local field. We know that for any $n \in \mathbb{N}$ there exists a unique ...
3 votes
0 answers
87 views

For any $m \in \mathbb{N}$, are there primes $p_1, \dots, p_n$ such that $\gcd(p_1 + 1, \dots, p_n + 1) = m$?

I've been thinking about this question and trying to apply Dirichlet's theorem on primes in arithmetic progressions, but can't seem to get the details of the argument. Dirichlet says there are ...
2 votes
1 answer
93 views

How do I compute the eleventh roots of unity algebraically, expressed with only square roots and fifth roots?

I was able to reduce the problem to solving the quintic equation $$x^5+x^4-4x^3-3x^2+3x+1=0,$$ which seems to be the minimal polynomial of $2\cos(2\pi/11)=e^{2\pi i/11}+e^{-2\pi i/11}$. But I couldn't ...
0 votes
0 answers
15 views

Exploring the Intersection of Expander Graphs, Number Theory, Representation Theory and Recent Computer Science Developments

I have a solid understanding of the basics of expander graphs and their properties and the recent development of High-Dimensional Expanders and their application to Random Walks, along with other ...
22 votes
2 answers
755 views

Strange behaviour of $x^2+5x+7$ under iteration

If any of the following exposition is unclear, please write a comment. In essence, I am looking at the graph $G$ that is generated by the polynomial $q(x) = x^2+ax+b$ ($a,b \in \mathbb{Z}$) via the ...
0 votes
3 answers
2k views

Prove that: $\gcd[a,b,c]=\frac{abc.\operatorname{lcm}(a,b,c)}{\operatorname{lcm}(a,b)\operatorname{lcm}(a,c)\operatorname{lcm}(b,c)}$

I may be wrong, but I was thinking of: $\operatorname{lcm}(a, b)$ as $\min(a, b)$ and $\gcd (a, b)$ as $\max(a, b)$ and $a <b <c$ I know I'm wrong, but I think I can do it
6 votes
3 answers
2k views

Find $x,y$ given $\gcd(x,y)$ and ${\rm lcm}(x,y)$

These two exercises I encountered recently seem to develop some type of connection between GCD and LCM I can't quite figure out. Exercise 1: Find all the numbers $x$ and $y$ such that: $a) \ GCD(x,y)=...
5 votes
2 answers
925 views

Difference between a Reduced Residue Class and a Reduced Residue System

Currently studying Dirichlets proof of Primes in arithmetic progressions. Is there a difference between a Reduced Residue Class and a Reduced Residue System? Any subset $R$ of the integers is called a ...
0 votes
2 answers
38 views

Finding any arbitrary integer point on a line with rational slope and intercept

Consider the equation of a simple line: $$ f(x)=mx+b $$ with the additional constraint that $m$ and $b$ are guaranteed to be rational numbers: $$ f(x) = \frac{m_n}{m_d}x + \frac{b_n}{b_d} $$$$ m_n, ...
6 votes
1 answer
229 views

A question about prime numbers, totient function $ \phi(n) $ and sum of divisors function $ \sigma(n) $

I noticed something interesting with the totient function $ \phi(n) $ and sum of divisors function $ \sigma(n) $ when $n > 1$. It seems than : $ \sigma(4n^2-1) \equiv 0 \pmod{\phi(2n^2)}$ only if ...
5 votes
1 answer
101 views

Algebraic points on circles/solutions to $x^2+y^2=a$ over $\overline{\mathbb Q}$: is there a point with odd degree field extension over $\mathbb Q$?

It is well known how to parameterize rational solutions to $x^2+y^2=a$ for $a\in \mathbb N^+$: first we know which $a$ admit integer solutions; our characterization is strong enough to tell us that ...
-3 votes
0 answers
80 views

Is there any way to tackle such questions? [closed]

ISI-2023-UG(B) Question (a) Let $n \geq 1$ be an integer. Prove that $X^n+Y^n+Z^n$ can be written as a polynomial with integer coefficients in the variables $\alpha=X+Y+Z$, $\beta=X Y+Y Z+Z X$ and $\...
0 votes
3 answers
47 views

Examples of integer sequences that have a distribution approx $1/\log(n)$, like the primes do?

It is well known that the primes are distributed such that they occur with an approximate "likelihood" of $1/\log(n)$ around the integer $n$ - or more precisely, the number of primes up to $...
1 vote
0 answers
79 views

Unique nonnegative solutions of linear diophantine equations in many variables

Given a positive integer $k$, fix $k$-many non-negative integers $n_1,\ldots, n_k\in \mathbb{N}$. Can you always find non-negative integer coefficients $a_1,\ldots, a_k\in \mathbb{N}$ so that the (...
3 votes
1 answer
29 views

Orbit of Lagrange resolvent under complex conjugation

The orbit of Lagrange resolvent $R = \sum{a_i w^i}$ under the action of $S_n$ plays a key role in Lagrange's method of solving polynomial equations. I've been playing with the case n=4 and noticed the ...
6 votes
1 answer
316 views

Why do so many solutions to $n+1\mid3^n+1$ satisfy $n\equiv27\pmod{72}$?

Here are the first $40$ integers for which $\dfrac{3^n+1}{n+1}$ is an integer. I've denoted their residue mod $72$ by color and a symbol. $$\begin{array}{r}\dagger\,\color{violet}0,&\bullet\,\...
2 votes
1 answer
64 views

Evaluation of nontrivial zeros of $\zeta$ in explicit formulae

Im sure this question is completely trivial, but I just want to check my understanding: For the various explicit formulae in Analytic Number Theory involving sums over the nontrivial zeros of $\zeta$ ...
3 votes
2 answers
146 views

Smallest prime number not divides $n$

For each integer $n>2$, we define $p(n)$ to be the smallest prime number that does not divide $n$. Prove that $$\lim_{n\to \infty}\frac{p(n)}{n}=0.$$ My argument is: We only need to prove $$\lim_{...
11 votes
3 answers
318 views

Is there a perfect square (other than 9) all of whose digits are 7, 8, or 9?

Clearly, $3^2=9$ is a perfect square, all of whose digits are $7$, $8$, or $9$. Are there any other perfect squares with this property? This is an interesting question that does not seem to be ...
1 vote
2 answers
118 views

Determine the number of elements of order $≤ 2$ in $(\mathbb{Z}/2^n\mathbb{Z})^×$

I have to determine the number of elements of order $≤ 2$ in $(\mathbb{Z}/2^n\mathbb{Z})^×$, and use this to find the rank and the elementary divisors of $(\mathbb{Z}/2^n\mathbb{Z})^×$ I know that $(\...
0 votes
0 answers
31 views

What is the meaning of logarithmic density in layman's terms?

The way that I understand natural density is that it is an "intuitive way of describing the size of a set", whereas cardinality is an "absolute way of describing the size of a set"....
0 votes
3 answers
213 views

Prove that there are infinitely many triples $(𝑎,𝑏,𝑐)$ of integers such that $a^2+b^2+c^2=a^3+b^3+c^3$

I have already made the observation that since the domain is real numbers, $a^2+b^2+c^2$ the three terms is always positive but $a^3+b^3+c^3$ since its to an odd power can have negative terms implying ...
30 votes
5 answers
5k views

When the product of dice rolls yields a square

Succinct Question: Suppose you roll a fair six-sided die $n$ times. What is the probability that the product of the rolls is a square? Context: I used this as one question in a course for elementary ...
3 votes
2 answers
178 views

Proof of $a^3 - b^3 = c^3 + d^3$, where $a,b,c,d$ all rational?

Reading Wikipedia article on Diophantus, it says in a book that survived that he makes reference to a lost book called Porisms and the theorem stated in the title: the difference between the cubes of ...
0 votes
0 answers
40 views

What does "almost all in the sense of logarithmic density" mean in Tao's Collatz paper?

Tao's paper: "Almost all orbits of the Collatz map attain almost bounded values" (via arXiv.org) 1st related question/discussion, I did not understand: Meaning of "almost all" in ...
0 votes
1 answer
2k views

Condition For 4th degree polynomial equation having positive roots

Consider the biquadratic polynomial equation $\rho_0y^4+\rho_1y^3+\rho_2y^2+\rho_3y+\rho_4y=0$, where $\rho_0,\rho_1,\rho_2, \rho_4$ are positive and $\rho_3$ is negative. So by Descartes' rule of ...
-1 votes
1 answer
116 views

Prove that no rational solution of equation $x^2-y^2=1002$ exist. [closed]

I have been able to prove in case of integers $x$ and $y$. My approach in case of rational numbers is as follows, Let $x=\frac{a_1}{b_1}$ and $y=\frac{a_2}{b_2}$ with $\gcd(a_1,b_1)=1$ and $\gcd(a_2,...
0 votes
0 answers
24 views

Interpolating polynomial for characteristic function of primitive Nth roots of unity among all Nth roots

In answering a question here, I had to make use of the unique polynomial $P_N(x) = \sum_{j=0}^{N-1} a_j x^j \in \mathbb{C}[x]$ whose value at any primitive $N^{th}$ root of unity is $1$, and whose ...
2 votes
1 answer
38 views

Dirichlet's approximation theorem with even or odd denominators

It follows from Dirichlet's approximation theorem that for any irrational $\alpha,\ 0<\left\lvert \frac{p}{q} - \alpha \right\rvert < \frac{1}{q^2} $ for infinitely many pairs of integers $(p,q)....
-1 votes
0 answers
95 views

Using Minkowski's Theorem to prove existence.

I need to use Minkowski’s theorem to show that if α ∈ R and Q ∈ N then there exist q, a ∈ Z such that 1 ≤ q ≤ Q and |qα − a| ≤ 1/Q. I believe that I need to define the lattice in R: Λ = {qα : 0<α&...
3 votes
1 answer
663 views

Is $\mathbb{Z}[\sqrt[3]{2}]$ a principal ideal domain? [closed]

Is $\mathbb{Z}[\sqrt[3]{2}]$ a principal ideal domain? That is, is every ideal of $\mathbb{Z}[\sqrt[3]{2}]$ generated by a single element?
5 votes
4 answers
19k views

The Diophantine equation $x^2 + 2 = y^3$ [duplicate]

How to solve the Diophantine equation $x^2 + 2 = y^3$ with $x,y>0$ ? ($x,y$ are integers.)
2 votes
1 answer
71 views

Computing Asymptotic Constant for the count of $S_3$-sextic fields

I am currently reading this paper counting $S_3$-sextic fields https://www.ams.org/journals/proc/2008-136-05/S0002-9939-07-09171-X/S0002-9939-07-09171-X.pdf by Bhargava and Wood. I'm trying to verify ...

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