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 algebra geometry, exponential and character sums, Zeta and L-functions, multiplicative and additive number theory, etc.

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19
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2answers
646 views

$\lim_\limits{x \to \infty} \frac1x \sum_\limits{n\leq x}\mu(n)=0 \iff$ Prime Number Theorem

I'm reading Analytic Number theory from Tom. M. Apostol's Introduction to Analytic Number Theory. In the fourth Chapter of the book he proves the equivalence of Prime number theorem with the ...
1
vote
0answers
56 views

Is $f$ multiplicative?

I was reading Apostol's Number Theory Book and I came across this question(I have rephrased it here for the sake of clarity): Let $f(\chi)$ denote the conductor of $\chi \mod k$ where $k = k_1 \cdots ...
2
votes
2answers
35 views

Understanding of real numbers based on Dedekind cut?

I can not understand the construction of real numbers by Dedekind cuts. Can somebody help me with understanding? The problem which I have is that the cardinality of rationals is $ \aleph_0 $. Base on ...
0
votes
0answers
22 views

Can't understand why 24 divides $m(mn+1)−n(m^2−1)=m+n.$ [duplicate]

I asked if If $mn+1$ is a multiple of $24$ , prove that $m+n$ is also a multiple of $24$. And i got this response: If $m$ is odd, then $8∣m^2−1=(m−1)(m+1)$ because $m−1,m+1$ are consecutive even ...
1
vote
1answer
646 views

Summation with Legendre Symbol $\sum_{a=1}^{p-1}\big(\frac{a}{p}\big)a=0$

I am trying to show the following: if $p$ is prime that is $1$ modulo 4, then $$\sum_{a=1}^{p-1}\bigg(\frac{a}{p}\bigg)a=0$$ I know I need to use the fact that for an odd prime p, $\frac{p−1}{2}$ of ...
1
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0answers
21 views

Is there a rational parameter for the cylindrical curve?

Is there a rational parameter for the cylindrical curve? \begin{align*} \left\{ \begin{split} x^2-y^2=2\\ y^2-z^2=3 \end{split} \right. \end{align*} It seems to have something to do with the topic of ...
0
votes
1answer
21 views

$\varphi(d)$ elements of order $d$ in $\mathbb{U}_n$ (defined in the question)

$$\mathbb{U}_{n\in\mathbb N^*}=\left\{\exp\left(i\frac{2\pi k}{n}\right),\,0\leq k\leq n-1\right\}$$ $$\varphi(d)=\text{card}\{k\in[[1,d]],\,\text{gcd}(k,d)=1\}$$ If $z\in\mathbb U_n$, we call order ...
30
votes
1answer
862 views

$2^n$th decimal place of $\sqrt{2}.$

Someone on Art of Problem Solving claims to know how to calculate the $2^{2020}$th decimal place of $\sqrt{2},$ and will tell us if everyone gives up. Brute force will not work, nor will a BBP style ...
0
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0answers
38 views

Evaluate $\zeta_F(0)$ where $F = \mathbb{Q}(\sqrt{-23})$

Using the modular forms database ([1], [2]) I could find the coefficients of the zeta function corresponding to the number field $F = \mathbb{Q}(\sqrt{-23}) \simeq \mathbb{Q}(x)/(x^2 - x +6)$. We ...
3
votes
1answer
86 views

In how many ways can a number be decomposed in a sum of distinct perfect squares

Is there a formula for how many ways can a number be decomposed in sum of distinct perfect squares. Example: $$50 = 49 + 1 = 1 + 4 + 9 + 36 = 9 + 16 + 25$$ The number 50 can be decomposed in 3 ...
3
votes
1answer
83 views

How do you define a non-integer as even or odd? [duplicate]

Note that in this question, I will use the word non-integers to mean all rational non-integers. I've been working on problems on non-integers lately, and a few thoughts popped into my head. How do ...
-1
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2answers
102 views

Show that rational numbers have the Archimedean property

By Archimedean property I mean: For any positive rational numbers $x = \frac{a}b$ and $y = \frac{c}d$, there is an integer $n$ such that $nx > y$, namely, $nx \equiv (x+x+ \ldots+x)$ with $n$ ...
1
vote
1answer
28 views

Manipulation of sums (in order to understand proof regarding Gauss sums)

I'm looking over the proof of $\left| \tau(\chi) \right|=\sqrt{q}$, when $\chi$ is a primitive character mod $q$, and $\tau$ the Gauss sum. (H. L. Montgomery and R. C. Vaughan. Multiplicative number ...
0
votes
0answers
20 views

Does completeness in $I$-adic topology implies $I$-adic completeness?

Let $R$ be a ring and $I\subset R$ be a principle ideal (so it is finitely generated). Suppose $R$ is complete with its $I$-adic topology. Does this implies $R$ is $I$-adically complete, i.e. $R\cong\...
1
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0answers
15 views

Does this set have Schnirelmann density $0$?

I am a bit confused about Schnirelmann densities in some cases. Recall that if $X$ denotes a subset of $\mathbb{N}$, then the Schnirelmann density of $X$ is defined as $$ \sigma X = \inf_n \frac{\pi(n)...
5
votes
3answers
2k views

Factor 43361 knowing $\phi(43361)$

It is given that the number $43361$ can be written as product of two distinct prime numbers $p_{1}$ and $p_{2}$. Further, assume that there are $42900$ numbers which are less than $43361$ and co-prime ...
0
votes
1answer
22 views

Linearly independent over $F_p$ to $Q$

Let $v_1, . . . , v_m$ be vectors with $n$ $(0,1)$-entries. Prove that if these vectors are linearly independent over $\mathbb{F}_p$ for some prime number $p$ then they are linearly independent over $\...
2
votes
1answer
74 views

Primes congruent to 1 modulo n

I was wondering whether the following statement is true, and if so, how to prove it: for all $n\in\mathbb{N}$, there exist prime numbers $p$ and $q$ such that $p\equiv1$ (mod $n$) and $q\equiv-1$ (...
0
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0answers
28 views

What about the algebraic structure of $T_p(G) \otimes_{\mathbb{Z}_p}A$?

Let $G$ be a $p$-divisible group over the ring of $p$-adic integers $O_K$ of $p$-adic field $K$. The $p$-adic Tate module $T_p(G)$ of $G$ is rank $1$ free $\mathbb{Z}_p$-module. Then $T_p(G) \otimes_{...
6
votes
1answer
165 views

How to prove ${}_{2}F_{1}\left(\frac{1}{3},\frac{2}{3};\frac{3}{2}; \frac{27}{4}z^2(1-z)\right) = \frac{1}{z}$

With reference to the following post On $\int_0^1\arctan\,_6F_5\left(\frac17,\frac27,\frac37,\frac47,\frac57,\frac67;\,\frac26,\frac36,\frac46,\frac56,\frac76;\frac{n}{6^6}\,x\right)\,dx$ I used ...
-1
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0answers
24 views

Show that $\mu* Id=Id*\mu$ where $*$ is the Drichlet convolution [closed]

I am trying to show that $\mu* Id=Id*\mu$ where $*$ is the Drichlet convolution $\mu$ is the mobius function and $Id$ is the identity function. I see it in my notes without proof and I do not see how ...
0
votes
1answer
44 views

What happens to point at infinity when you complete the square of an elliptic curve?

Suppose we have some elliptic curve (just suppose we are working over $\mathbb{R}$) defined by the equation $$Y^2Z + a_1XYZ + a_3YZ^2 = X^3 + a_2X^2Z + a_4XZ^2 + a_6Z^3 $$ with a point at infinity $[0,...
0
votes
0answers
17 views

Factor-square property [duplicate]

Polynomial f(x) has the factor-square property if f(x) is a factor of f($x^2$). List all of the monic polynomials of degree 1, of degree 2, of degree 3, and of degree 4. Are there monic factor-square ...
1
vote
1answer
28 views

Positive Totient function?

We know that: $$\phi(n) = n(1- 1/p_1)(1-1/p_2)\ldots(1 - 1/p_k)$$ where $\phi$ is the Euler totient function and $p_i$ are the primes dividing $n$. Is anything known about the similar function $$\...
3
votes
2answers
122 views

Is the following definition of an elliptic curve correct?

Im new to algebraic geometry so I want to make sure im getting my definitions right. I know there are a few ways to state what an elliptic curve is (ex a smooth projective curve of genus one with ...
0
votes
1answer
20 views

What the probabilty of selecting different congruences of numbers?

I was reading through an article the other day regarding the probability of selecting relatively prime integers, which is known to be $\frac{6}{\pi^2}$. How would you compute the probability of ...
1
vote
2answers
139 views

Proof verification of non vanishing of $ ~L(1, \chi) \neq 0~$ for real valued character

I am self studying analytic number theory from Tom M Apostol introduction to analytic number theory and I am asking for solution verification for a part of Theorem 6.20 of Apostol. I am adding it's ...
0
votes
0answers
102 views

How to solve polynomial equations of more than one transcendental functions in closed form?

Let $f_1,f_2$ functions in $\mathbb{C}$ so that $f_1,f_2$ are algebraically independent, $P$ an irreducible bivariate polynomial function in $\mathbb{C}$ whose coefficients are algebraic numbers. Why ...
3
votes
1answer
637 views

How to prove $\sum_{ d \mid n} \mu(d)f(d)=\prod_{i=1}^r (1-f(p_i))$?

I have to prove for $n \in \mathbb{N}>1$ with $n=\prod \limits_{i=1}^r p_i^{e_i}$. $f$ is a multiplicative function with $f(1)=1$: $$\sum_{ d \mid n} \mu(d)f(d)=\prod_{i=1}^r (1-f(p_i))$$ How I ...
0
votes
0answers
21 views

What are some elementary bounds for Mertens' Third Theorem?

In particular, I am looking for $A$ and $B$ such that, for all $x>s$, $$\frac{e^{-\gamma}}{\log x} B< \prod_{ p \leq x} \frac{p-1}{p}<\frac{e^{-\gamma}}{\log x}A.$$ Rosser provides both ...
1
vote
1answer
32 views

Converting between different ways of representing integer partitions

I'm aware of two ways of representing integer partitions, and I'm trying to understand how to write the mapping between then in terms of mathematical notation. Firstly, we can say that an integer ...
0
votes
3answers
58 views

Is there a Prime number test that has no false positives

The primality tests that I have found, i.e. Fermat and Miller-Rabin don't return any false negatives, but do sometimes return false positives. That is, some composites are incorrectly decided as ...
3
votes
1answer
39 views

coefficient $[q^n]\sum_{m\ge1}\frac{q^{\alpha m}}{1-q^{\beta m}}$ where $0<\alpha<\beta$

Problem: I am looking for a finite-sum expression for the coefficient $c_n=c_n(\alpha,\beta)$, where $$C(\alpha,\beta;q)=\sum_{m\ge1}\frac{q^{\alpha m}}{1-q^{\beta m}}=\sum_{n\ge1}c_n(\alpha,\beta)q^n,...
0
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0answers
18 views

Finding bias with n-bit number representations given max and min

I'm a bit stuck on this question (not sure if it comes under math but if it doesn't please help me find which topic to put it in) : If you want an n bit number to have a max value of x (positive) and ...
0
votes
0answers
39 views

How to get $a_n$ if we have the following summation satisfied? [closed]

$$\frac{1}{a_1a_2a_3}+\frac{1}{a_2a_3a_4}+\cdots+\frac{1}{a_n a_{n+1}a_{n+2}}=(n+3)\frac{a_n}{4a_{n+1}a_{n+2}}$$
0
votes
1answer
25 views

If two multiplicative functions are close, are their Dirichlet inverses necessarily close?

Let $f$ be a multiplicative arithmetic function such that $\sum_{n = 1}^{\infty} |f(n)|$ converges. Given some $\epsilon > 0$, can we necessarily choose some $\delta$ such that for all arithmetic ...
2
votes
2answers
77 views

Seeking upper bound on largest prime factor of colossally abundant numbers

I recently came across an older post where the OP referred to papers giving upper bounds on the size of the largest prime factor in superabundant and colossally abundant numbers. I asked for links to ...
5
votes
4answers
42k views

What is the remainder when 4 to the power 1000 is divided by 7

What is the remainder when $4^{1000}$ is divided by 7? In my book the problem is solved, but I am unable to understand the approach. Please help me understand - Solution - To find the ...
0
votes
0answers
86 views

Does this product of ratios always result in an integer?

Claim: the product: $$ \prod_{(i,j)\in [N]^2, i\neq j} \dfrac{m_j-m_i}{j-i}$$ is an integer if the set $\{m_k\}$ are all integers, where $1 \le k\le N$ (i.e. $k\in [N]$). And if not please give a ...
4
votes
2answers
108 views

Find the integer solutions to $4^x - 9^y = 55$

I want to find the integer solutions of: $$ 4^x - 9^y = 55$$ For now, I see that $x = 3, y = 1$ is an integer solution to the equation. How can I rigorously prove there are no other solutions for $x, ...
1
vote
0answers
17 views

$L$-functions in $q$ aspect

When we are interested in the $L$-functions in $q$ aspect, how do we start? Like for the $t$ aspect (or level aspect), we integrate over $t\asymp T$. If we sum over the conductor $q\asymp Q$ then do ...
8
votes
3answers
146 views

Is $(3^p-1)/2$ always squarefree?

I have little conjecture. Maybe it's stupid i don't know. Let $p>5$ be a prime number. Then $(3^p-1)/2$ is always squarefree? It's true for $p<192$.(I used Mathematica.)
2
votes
3answers
98 views

Does the sum of the power of the smallest prime factor grow linearly?

Let $f(p_1^{a_1}...p_k^{a_k})=a_1$ where $p_1<...<p_k$ and let $g(n)=\sum_{k<=n}f(k)$. Then is $g(n)=\mathcal{O}(n)$? I know that for $f(p_1^{a_1}...p_k^{a_k})=k$ we would have gotten $g(n)=\...
0
votes
0answers
18 views

Criterion for an extension of complete fields to be unramified

Let $A$ be a complete $DVR$ with fraction field $K$. Let $L/K$ be a finite separable extension of $K$ and let $B$ be the integral closure of $A$ in $L$. Then $B$ is also a complete $DVR$ and let us ...
1
vote
1answer
50 views

Using quadratic residue

show that $2b^2+ 3$ cannot be a divisor of $a^2-2$ I have tried working with residue of $2$ modulo $p$ where $p$ is a divisor of $2b^2+3$ but couldn't stablish the result.
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votes
0answers
32 views

Determine the remainder of p1 + p2 + p3 + … + p2021 when divided by 16 [closed]

For any natural number n, it is defined as the smallest prime factor of n^8 + 1. Determine the remainder of p1 + p2 + p3 + ... + p2021 when divided by 16. What should i do? I need help to work this ...
1
vote
1answer
41 views

Prime zeta function and squarefree divisors

Let $x_n$ and $y_n$ be two random numbers, drawn uniformly and independently from $\{1,\ldots,n\}$. Write $d_{\text{sf}}(x_n,y_n)$ for the number of squarefree divisors of $\gcd(x_n,y_n)$. We are ...
-2
votes
0answers
48 views

Importance of Riemanns paper [closed]

Riemann found a connection between complex analysis and number theory and found an exact formula for $\pi(x)$. As we can see in this question we have a way easier formula for the prime counting ...
3
votes
1answer
63 views

Using Quadratic Residue to show that $2^m-1$ doesn't divide $3^n-1$

Show that if $m$ is congruent to $n$ modulo $2$ and $m>1$ then $2^m-1$ cannot be a divisor of $3^n-1$ What I have tried: I showed that if $p$ is a divisor of $2^m-1$ then if $m$ and $n$ are odd ...
4
votes
1answer
4k views

Number of squares crossed by a diagonal

How many boxes are crossed by a diagonal in a rectangular table formed by $199 * 991$ small squares ? Well, the answer is $a+b-\gcd(a, b)$. but, How can I prove that? I tried drawing different ...

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