Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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5
votes
1answer
117 views

Solutions of a cubic diophantine equation in $\mathbb{Z}/p\mathbb{Z}$

Suppose $p\in\mathbb{Z}$ is prime and $p\equiv 1\pmod{3}$. Is there an estimate of the number of solutions of $x^3+y^3=z^3$ in $\mathbb{Z}/p\mathbb{Z}$, preferably using elementary number theory and ...
0
votes
1answer
88 views

Divisors of $q^kp^r$

This is a generalization of my previous problem. Let $p$ and $q$ be prime numbers. What is the necessary and sufficient condition (in terms of $p,q$ and $k,r$) such that we can partition the divisors ...
6
votes
3answers
143 views

Determine the minimum of $a^2 + b^2$ if $a,b\in\mathbb{R}$ are such that $x^4 + ax^3 + bx^2 + ax + 1 = 0$ has at least one real solution

I just wanted the solution, a hint or a start to the following question. Determine the minimum of $a^2 + b^2$ if $a$ and $b$ are real numbers for which the equation $$x^4 + ax^3 + bx^2 + ax + 1 = 0$$ ...
1
vote
2answers
76 views

Integer outputs of $y=x^2$ , do their last digits form an irrational?

Let the domain of $y=x^2$ be the positive integers. I input consecutive positive integers from $[1, \infty)$ their last digits are $a, b, c, ...$ respectively. If I then make the number $z=\frac ...
6
votes
1answer
136 views

What is the smallest real $q$ such that there is always a prime between $n^q$ and $(n+1)^q?$

In this answer, it is mentioned that for $q=3$, we are guaranteed the existence of a prime between $n^q$ and $(n+1)^q$, and that it is conjectured that this is true for $q=2$. I am wondering though, ...
2
votes
1answer
62 views

If $p$ is a prime number, prove that for any $a \in \mathbb{Z}$, we have $p |a^p+(p-1)!a$ and $p|(p-1)!a^p+a$

If $p$ is a prime number, prove that for any $a \in \mathbb{Z}$, we have $$p |a^p+(p-1)!a$$ and $$p|(p-1)!a^p+a$$ I totally got no idea how to start. Can anyone give some hints?
3
votes
1answer
111 views

Krull's intersection theorem in the q-expansion principle

I'm currently reading the proof of the q-expansion principle in Katz'73 paper "p-adic properties of modular schemes and modular forms" . The principle itself is a Corollary (1.6.2) of Theorem 1.6.1, ...
3
votes
1answer
115 views

Is there any kind of irrational number wich does not contain digit 9?

At first we must prove that there is or is`t irrational numbers which does not contain digit 9! if there are many kind of such numbers, then there is another question: how to write down algebraic ...
0
votes
3answers
337 views

how to find out any digit of any irrational number?

We know that irrational number has not periodic digits of finite number as rational number. All this means that we can find out which digit exist in any position of rational number. But what about ...
4
votes
2answers
638 views

How prove this $[\sqrt{23n}]\{\sqrt{23n}\}>3$

show that for any positive integer $n\ne 23m^2,m\in N$, have $$[\sqrt{23n}]\{\sqrt{23n}\}>3$$ and $\{x\}=x-[x]$ I have post this How prove this $|\{n\sqrt{3}\}-\{n\sqrt{2}\}|>\frac{1}{20n^3}$ ...
2
votes
1answer
44 views

Differentiating between prime/semi-prime and other integers

Does there exist a test that checks if a number is prime or a semi prime in polynomial time? I am aware that AKS can be used to check primality but what about semi primality? ...
12
votes
1answer
168 views

What is the significance of the power of $3$ in the sequence of primes given by $\lfloor A^{3^n}\rfloor ?$

Mill's constant is a number such that $\lfloor A^{3^n}\rfloor$ is prime for all $n$. The existence of such an $A$ was proven in $1947$. I know little about number theory, but I am curious as to why ...
1
vote
4answers
415 views

Rational solutions for $x^2+y^2=3$

Are there any rational numbers $x$ and $y$ such that $x^2+y^2=3$. I think there are no rational solutions, but I haven't been able to prove it.
1
vote
0answers
68 views

a diophantine equation from Stewart and Tall [duplicate]

This is from Stewart and Tall from the chapter on Kummer's Theorem. Show that there are no non trivial (non-zero) solutions to $x^3 + y^3=3z^3$
3
votes
1answer
545 views

Characterization of integers which has a $2$-adic square root

Does anyone know an "elementary" proof of the following theorem? Let $k \neq 0$ be a rational integer. Then $k$ admits a square root in $\mathbb{Z}_2$ if $k = 4^a (8b+1)$ for some $a \in \mathbb{N}$, ...
2
votes
0answers
48 views

Number of primes of type 4*n +1 in a range

I want to find number of primes which are congruent 1 (mod 4) in a range [a, b]. The range can be of order $10^9$ as a and b can be from $1$ to $10^9$. I tried segmented sieve but for a range so ...
8
votes
2answers
181 views

Solve $x^2+y^2=2$ for $x,y\in\mathbb Q$.

Solve $x^2+y^2=2$ for $x,y\in\mathbb Q$. I think the answer should be in terms of 1 integer variable $\in\mathbb Z$ only. I rewrite the equation to $(x+y)^2+(x-y)^2=2^2$, then by the formula of ...
1
vote
0answers
72 views

For squarefree $i$ what is $\sum_1^{n} \frac{1}{i}$?

For squarefree $i$ what is $\sum_1^{n} \frac{1}{i}$ ? I use $\sum_{\sqrt{n}>m>1} \mu(m) ln(\frac{n}{m^2}+\frac{1}{2})$. I know about the connection with $\zeta(2)$ and ...
2
votes
0answers
165 views

Approximation to $\pi(x)$ conjecture.

A friend conjectured that $\left[\prod_{k=1}^{a_j <\sqrt{x}} \left(1-\frac{1}{a_k}\right)\right] x$ is usually closer to $\pi(x)$ than $\operatorname{Li}(x)$ is for some (fixed) sequence of ...
4
votes
1answer
779 views

The Diophantine equation $x^2 + 2 = y^3$

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

This correct this demonstration of Number theory (binomial Expressions)

$$\\$$Em minha apostila tem as demonstrações dos seguintes lemas:$$\text{Lema (*): Sejam $a,m,n,q,r\in\mathbb{N}$ com $a\geq2$ tais que $m=nq+r$ then:}\\(a^m-1,a^n+1)=\begin{cases}(a^n+1,a^r-1)& ...
0
votes
2answers
69 views

Number obtained by reversing digits. Find the value satisfying a condition.

Let $a$ and $b$ be two-digit integers such that $b$ is obtained by reversing the digits of $a$. The integers $a$ and $b$ satisfy $a^2-b^2=m^2$ for some positive integer $m$. Which could be value of ...
1
vote
4answers
71 views

Determining the general form of $10^x \bmod 210$

While solving a problem I came across solving $10^x\bmod 210$ for various values of $x$. It seems that the values repeat after an interval of 6 for $x\geq4$. Can any one explain how can solve this ...
2
votes
0answers
42 views

Discriminant of $\mathrm{Tr}(xy)$ is $p^2$ in a maximal order of discriminant p in a quaternion algebra

I am trying to prove that $\mathrm{Tr}(xy)$ has discriminant $p^2$ in a maximal order of discriminant $p$ in a quaternion algebra. Does anybody have any hints to get a handle on this?
8
votes
3answers
176 views

Prove that $\lceil(\sqrt{3}+1)^{2n}\rceil$ is divisible by $2^{n+1}$.

Let $n$ be a positive integer. Prove that $\lceil(\sqrt{3}+1)^{2n}\rceil$ is divisible by $2^{n+1}$. I tried rewriting $\lceil(\sqrt{3}+1)^{2n}\rceil$ as $m*2^{n+1}$ for some m, but couldn't get ...
1
vote
1answer
108 views

Lower bound on the rank of the elliptic curve $y^{2} = x^{3} + A x^{2} + B x$

A recent post on Math.SE discusses an upper bound on the rank in terms of the number of distinct prime divisors of $A$ and $B$, namely, \begin{align} r \leqslant \omega(A^{2} - 4 B) + \omega(B) - 1, ...
2
votes
1answer
77 views

Group of finite ideles

A simple question: If $\overline{\mathbb{Q}}$ denotes the algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$ then what is the definition of the group of finite ideles of $\overline{\mathbb{Q}}$? ...
3
votes
1answer
151 views

Is there an explicit formula connected to $(\log\zeta(s))^2$?

Riemann used $\log\zeta(s)$ and, essentially, Perron's formula to find the explicit formula for his prime counting function, $\Pi(n)$: $li(x)-\displaystyle\sum_{\rho}li(x^\rho)-\log ...
4
votes
4answers
225 views

Showing limit of a sequence $0, \frac12, \frac14, \frac38, \frac5{16}, \frac{11}{32}, \frac{21}{64},…$

How do you show the convergence of the following 2 sequences? $0, \dfrac12, \dfrac14, \dfrac38, \dfrac5{16}, \dfrac{11}{32}, \dfrac{21}{64},...$ and $1, \dfrac12, \dfrac34, \dfrac58, ...
4
votes
2answers
256 views

Roots of monic polynomial over a number ring

If $R$ is a number ring with number field $K$ and $f$ is a monic polynomial over $R$, then I want to show that any root of $f$ is an algebraic integer.
6
votes
1answer
270 views

Let $F_n$ be a Fibonacci number and $p$ a prime. Verify that for $p \le 61$, if $p\equiv\pm1 \pmod{5}$ then $p\mid F_{p-1}$

Define the Fibonacci entry point of $p$ to be the least integer $n$ such that $p\mid F_n$ So for example, for $p = 3$ - the Fibonacci entry point is $n = 4$ since $F_4 = 3$ and obviously $3\mid 3$. ...
0
votes
4answers
1k views

$\sqrt{17}$ is irrational: the Well-ordering Principle

Prove that $\sqrt{17}$ is irrational by using the Well-ordering property of the natural numbers. I've been trying to figure out how to go about doing this but I haven't been able to.
7
votes
1answer
202 views

Is $\sum_{k=0}^{\infty}\frac1{2^{k^2}}$ rational? Transcendental?

Is $\sum_{k=0}^{\infty}\frac1{2^{k^2}}$ rational? Clearly this series is convergent (compare to geometric series with ratio 1/2). I'm sure it's irrational since a rational number written in base 2 ...
1
vote
1answer
186 views

how do we know the BBP formula for $\pi$ is valid?

I recently read about the Bailey–Borwein–Plouffe formula for calculating the $n^{\rm th}$ digit of $\pi$. I'm curious to how can we be sure that the formula is always accurate or correct?! Even if we ...
2
votes
2answers
89 views

Question related to trace function

I was reading this document called Discriminants and Ramified Primes. Suppose $O_K$ is the ring of algebraic integers of a number field $K$. Let $\mathcal{P}$ be a prime ideal of $O_K$ above $p$ such ...
3
votes
1answer
113 views

Inequality related to the continued fraction expansion of sqrt(3)

I am working on a problem related to the continued fraction expansion of $\sqrt3$. If $p_k$ and $q_k$ denote the numerator and denominator, respectively, of the $k$th convergent, I should show that ...
8
votes
4answers
592 views

The asymptotic expansion for the weighted sum of divisors $\sum_{n\leq x} \frac{d(n)}{n}$

I am trying to solve a problem about the divisor function. Let us call $d(n)$ the classical divisor function, i.e. $d(n)=\sum_{d|n}$ is the number of divisors of the integer $n$. It is well known that ...
3
votes
1answer
289 views

On an exponential diophantine equation

I am trying to find all integer solutions of $5^x + 12 ^y$ = $13^z$. The obvious (and pursued) solution is $(2, 2, 2)$, and no others. I've tried to use an appropriate modular arithmetic, but to no ...
0
votes
2answers
90 views

Primes corresponding to embeddings of a number field

Let $k$ be a number field. Define a prime of $k$ to be an equivalence class of absolute values on $k$. If $\sigma:k\hookrightarrow \mathbb{C}$ is an embedding of $k$ into the complex numbers then we ...
10
votes
2answers
135 views

An irreducible $f\in \mathbb{Z}[x]$, whose image in every $(\mathbb{Z}/p\mathbb{Z})[x]$ has a root?

Problem: Is there an irreducible $f\in \mathbb{Z}[x]$, whose image in every $(\mathbb{Z}/p\mathbb{Z})[x]$ has a root for $p$ prime? If there is, what is the minimal degree possible? I can only ...
5
votes
5answers
274 views

Whether the map $x\mapsto x^3$ in a finite field is bijective

Suppose $p\in\mathbb{Z}$ is prime and $\mathbb{F}_p:=\mathbb{Z}/p\mathbb{Z}$ is the finite field of size $p$. Now, consider the map: $$f:\mathbb{F}_p \to \mathbb{F}_p$$ given by $f(x)=x^3$. Then, (1) ...
6
votes
1answer
366 views

A general explicit formula for the generalized divisor summatory function?

Mertens function has, by residues, an explicit formula of $M(x)=\displaystyle\sum_{\rho}\frac{x^\rho}{\rho\zeta'(\rho)}-2+\sum_{n=1}^\infty\frac{(-1)^{2 n}(2\pi)^{2n}}{(2n)! n \zeta(2n+1)x^{2n}}$ ...
2
votes
1answer
37 views

Finite set of congruences

Is it true that for every $c$ there is a finite set of congruences $a_i(mod\,\,n_i) , c = n_1<n_2<n_3<...........<n_k \,\,\, (1)\\ $ So that every integer satisfies at least one of ...
3
votes
2answers
247 views

Show that 3 divides $\sigma(3n+2)$

I would like to show that for any integer $n \geq 0$, $3|\sigma(3n+2)$, where $\sigma(n)$ denotes the sum-of-divisors function. I have been able to show some results, for example that if $(3n+2)$ has ...
3
votes
2answers
443 views

Finding modulus by composite number

Let $p_1$ and $p_2$ are prime numbers, $a,b <p_1p_2$. I need to find $ab \pmod{p_1p_2}$. But for some reason I cant find $ab$, it will be too large number. How can I find it without calculating ...
2
votes
1answer
70 views

Integral elements of a non-archimedean completion of a number field

Let $k$ be a number field, $v$ a discrete non-archimedean valuation on $k$. Let $k_v$ be the completion of $k$ wrt $v$ and $\mathcal O_v$ its valuation ring. My question is: If $x\in k_v$ then is ...
4
votes
0answers
122 views

Prove That $1/\sin(i \pi/n)+1/\sin(j \pi/n) \ne 2$ When $n$ is Odd

The original problem is: Prove that $1/\sin(i \pi/n)+1/\sin(j \pi/n) \ne 2$ when $n$ is odd. I tried, and found that although everything looks similar, I actually know nothing about this kind of ...
0
votes
1answer
1k views

What better way to check if a number is a perfect power?

What better way to check if a number is a perfect power? Need to write an algorithm to check if $ n = a^b $ to $ b > 1 $. There is a mathematical formula or function to calculate this? I do not ...
1
vote
1answer
198 views

Linear equation with prime coefficient.

Suppose we have a linear equation with two variables say $x$ and $y$ and three integer coefficient $a , b$ and $c$ (constant), where $a$ and $b$ are prime all are greater than zero. $ax+by=c$ how ...
4
votes
0answers
181 views

Proof of $\pi$ not being a quadratic irrational number.

Does someone know a proof (books , articles) that $\pi$ is not a quadratic irrational? The proof should not use that $\pi$ is transcendental. Any hints would be appreciated.