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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Harmonic Distribution of prime numbers

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 ...
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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-...
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  • 9,550
1 vote
1 answer
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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 ...
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4 votes
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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{...
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7 votes
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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{...
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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 ...
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  • 2,147
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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 ...
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-1 votes
1 answer
22 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/...
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  • 2,147
-1 votes
1 answer
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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 ...
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-3 votes
0 answers
30 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 ...
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  • 2,147
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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 ...
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2 votes
1 answer
47 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\...
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-1 votes
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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 ...
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  • 2,147
0 votes
2 answers
39 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 ...
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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 ...
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Primitive root and prime p such that p' =4p +1 is also a prime

The following question is from my assignment in number theory and I am not able to make any progress on this. Question: If p is a prime of the form p=4p'+1 where p' is also a prime then 2 is a ...
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1 vote
1 answer
35 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/...
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1 vote
1 answer
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$M_k(\Gamma_1(N)) = \bigoplus_{\chi \mod N} M_k(N, \chi)$

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\...
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1 vote
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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 $...
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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 ...
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2 votes
0 answers
47 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 ...
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5 votes
1 answer
60 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^{...
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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) ...
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  • 2,147
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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/~...
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0 answers
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Questions in proof of Arithmetic Large Sieve

I am studying Arithmetic Large Sieve from following notes of Zeev Rudnick:http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html I have questions in lecture 14 here: http://www.math.tau.ac.il/~...
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  • 2,147
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Change of fractional numbers from one base to another

Suppose $x=0.a_1a_2\cdots a_n a_{n+1} \cdots$ is a nonterminating fraction in base $b>1.$We want to find the first n digits in base 10 accurately.Could somebody kindly tell how we can do it.I ...
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8 votes
0 answers
73 views

"Multiply everything so far, plug into polynomial" - can these always yield primes?

Say that a factonomial sequence is a (possibly infinite) sequence of natural numbers $x_i$ such that each $x_i$ is prime, and there is some (single variable, integer coefficients, nonconstant) ...
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24 views

Classifying all binary quadratic forms over Z of a positive discriminant D

My textbook has a method for finding what quadratic forms are possible given a discriminant but that's only for positive definite binary quadratic forms. For example if the discriminant is $-100$ then ...
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0 answers
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Natural density of sets using Lebesgue measure

Suppose that the sets $A$ and $B$ are specified subsets of positive integers up to $n$. (For instance, $A$ or $B$ could be the set of all even integers less than $n$). Assume also that $A$ and $B$ ...
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4 votes
2 answers
126 views

Periodic sequences of integers generated by $a_{n+1}=\operatorname{rad}(a_{n})+\operatorname{rad}(a_{n-1})$

Let's define the radical of the positive integer $n$ as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\ p\text{ prime}}}p$$ and consider the following Fibonacci-like sequence $$a_{n+1}=\...
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1 answer
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A question in proof of analytic large sieve

I have been reading Sieve theory from notes of Zeev Rudnick here:http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html and I have a question on page 5 of lecture 14 here: http://www.math.tau.ac....
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A question in proof of Linnik's Theorem in Arithmetic Large Sieve

This question is from course notes in sieve theory and I am struck on this assertion in the proof of Linnik's theorem. Consider Page 4 of lecture 14 here: http://www.math.tau.ac.il/~rudnick/courses/...
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How to estimate S(z) in Arithmetic Large Sieve

This question is part of a proof in course in Sieve Theory( http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html, precisely lecture 11 and 14)and I am not able to prove this particular ...
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1 vote
0 answers
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$\sum_{ n \leq N , p | n => p < N^{\epsilon} } 1 \gg_{\epsilon} N.$

I am reading class notes of number theory of a senior and I am struck an assertion of the proof. I have thought about it many times, so I am posting it here. An integer n is called Y-smooth if $p|n =&...
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-1 votes
0 answers
38 views

How to prove that gcd( d,q )=1

This question is from my class notes in Number Theory and I am not able to prove this deduction in a proof. Proof ( I am adding only the part of the proof relevant to this question): Let $$A= \{ n \...
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0 votes
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How to prove that Vinogradow inequality implies that...

This question was asked in my assignment on Number Theory and I am struck on it. Define $n_p$ = min { $ 1 \leq m \leq \frac{(p+1) } {2} : ( m/p)=-1$}. Assuming polya vinogradow inequality prove ...
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1 answer
35 views

Notation $(S,*)$ or $(\mathbb S,*)$ for semigroup leading to group $(\mathbb G,*)$

In a number-theoretic context, is it best to use the notation $(S,*)$ or $(\mathbb S,*)$ for a generic semigroup leading to group $(\mathbb G,*)$ by taking equivalence classes under an equivalence ...
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0 votes
2 answers
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How to rigorously interpret and transform "equal chance" in different ways?

Put $100$ identical balls into $10$ identical boxes in a way that each ball enters each box with an equal chance. What's the probability that no box is empty? I have solved it but like to discuss ...
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2 votes
1 answer
45 views

Field over which CM endomorphism is defined

Let $E$ be an elliptic curve with coefficients over some number field $K$. Is it true that if $E$ has complex multiplication by $\mathbb{Q}[\sqrt{-D}]$, then any endomorphism $\phi: E \rightarrow E$ ...
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1 vote
2 answers
63 views

Show that there does not exist integers $n_{1}, \ldots, n_{8}$, not necessarily distinct, such that $n_{1}^{4}+\cdots+n_{8}^{4}=1993$.

I tried using mod of $8$ and got that mod $8$ of an even number raised to the power $4$ is $0$, and $1$ in case of odd number. Also, $1993$ is $1$ mod $8$, but can't make progress from this point. ...
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0 votes
0 answers
18 views

Signature of a permutation couple

I was working on a problem when I found out I needed to know the signature of a permutation of the form : \begin{equation} (\sigma_A,\sigma_B) \end{equation} meaning that $(\sigma_A,\sigma_B)$ is ...
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  • 441
1 vote
1 answer
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Units in ring of integers of $5^{th}$ cyclotomic field

Let $K=\mathbb{Q}(\omega)$ with $\omega$ a primitive $5^{\text{th}}$ root of unity. I'm trying to prove that $$ \mathcal{O}_K^\times=\left\{\pm\omega^{a}\left(\frac{1}2+\frac{\sqrt{5}}2\right)^b: a,b\...
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0 answers
78 views

Intuition behind $\frac{a}{b} \equiv k \pmod{p} $

I am working with $p$-adic numbers at the moment and am having some trouble with a basic fact. I know that for $\frac{a}{b}\in\mathbb{Q}$ there is a solution $k\in\mathbb{Z}$ to $\frac{a}{b} \equiv k\...
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1 vote
0 answers
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Quadratic forms: Existence of $(x,y)\in \mathbb{Z}^2 \setminus \{0\}$ such that $P(x,y) < 2\sqrt{\lvert \det(P) \rvert}$

I am stuck on the following exercise: Show that for any non-degenerate quadratic form $P$ over $\mathbb{R}$, that is either indefinite or positive definite, exists an integer point $(x,y) \in \mathbb{...
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  • 2,452
0 votes
1 answer
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Proving that this infinite product is convergent

This question was asked in my assignment in number theory and I could not prove it. Question : Define the multiplicative function w(n) such that $w(p^k) =0$ for $k\geq 2$ and w(p)= { $\frac{p} { f(p)...
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0 answers
23 views

Intersection of meaning between Lebesque Measure and Natural Density

FYI: the articles linked and their content within are well-known in number theory, so number theory is a tag (correct if need be). $\textbf{Background}$: This article by Maier states that a given ...
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-3 votes
0 answers
27 views

Binary quadratic forms over Z of discriminant D [closed]

For each of the following 3 discriminant D = 4d, how can I classify all binary quadratic forms over Z of discriminant D. Is there a way to describe the river corresponding to the canonical form Q(x, y)...
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0 votes
0 answers
17 views

Number Theory , Primes and sum of squares [duplicate]

Show that if $a^2+b^2≡0 \pmod{p}$ , with $p$ a prime number and $p≡3\pmod{4}$ Then automatically $a≡0\pmod{p}$ and $b≡0\pmod{p}$ What I have done so far, Suppose $a^2≡0\pmod{p}$ and $b^2≡0\pmod{p}$ , ...
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-1 votes
0 answers
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Equivalence classes of positive definite binary quadratic forms [closed]

Find all equivalence classes of positive definite binary quadratic forms over $\mathbb{Z}$ with discriminant $D = −164$ under the action of the group $SL_2(\mathbb{Z})$. Can you find the ...
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-1 votes
0 answers
17 views

Applications of Diophantine equations? [duplicate]

It had proved that there is no algorithm to solve Diophantine equations, for that reason I want to know what are the Diophantine equations that physicists or chemists need to solve? or any other ...
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