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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3
votes
3answers
697 views

How to appreciate Fermat's last theorem?

I am someone who is not a Maths major, these days (during the summer) I am attracted to Fermat's Last Theorem. I understand that there is no whole number solution to the equation $x^n + y^n = z^n$ for ...
2
votes
1answer
34 views

Counting Coprime Numbers in a range:

I know that $\varphi(n)$ is the number of positive integers less than $n$ that are coprime to $n$. What I don't know is how to solve a related, but seemingly reverse problem. How do I count the ...
4
votes
1answer
51 views

Product of two sets with density zero has density zero?

Let $A$ and $B$ be two subsets of $\mathbb N$ which have asymptotic density zero. Define $A\times B$ as the set of integers of the form $ab$ with $a\in A$ and $b\in B$. Must $A \times B$ also have ...
2
votes
1answer
15 views

Bound on Lynden words made of $q$ letters

Let $N(q,n)=\frac{1}{n}\sum_{d|n}\mu(n/d)q^d$ for $q$ positive integer. Is it true that $N(q,n)<q^n/n$? This is true for $q$ prime which corresponds to the number of monic irreducible polynomials ...
3
votes
0answers
62 views

Zariski density of points over completion

I have a simple question which I couldn't find a reference to. Let $X$ be a smooth projective irreducible variety over $\mathbb{Q}$. Suppose we base change to $\mathbb{Q}_p$ (the $p$-adics) and ...
-4
votes
0answers
28 views

Don't exist $P(x)\in \mathbb{Z}[x] $ so that $P(x)$ is prime for all $x\in\mathbb{Z}$. [on hold]

I need show that don't exist $P(x)\in \mathbb{Z}[x] $ so that $P(x)$ is prime for all $x\in\mathbb{Z}$.
0
votes
2answers
19 views

What is an algorithm for describing the partition of this equivalence relation?

Let $\mathbb{R}$ be the set of real numbers,$ f : \mathbb{R} \to \mathbb{R}$ a map, and $E$ the equivalence relation on $ ℝ $ defined by $E = \{(x,y) \in \mathbb{R} \times \mathbb{R} \mid ...
0
votes
0answers
18 views

Selberg combinatorial identity

I am reading Granville's article on bounded prime gaps and in Section 4.5, he says that suppose $L(d)$ and $Y(r)$ are sequences of numbers supported only on the square-free integers. If $$Y(r) := ...
4
votes
1answer
52 views

About Mertens' first theorem

Mertens first theorem states that $ \sum_{ p \le x } \frac{\log p}{p} = \log x + R $ with $| R | \le 2$ . Is it correct that the limit $ \lim_{x \to \infty} \sum_{ p \le x } \frac{\log p}{p} - \log x ...
3
votes
1answer
45 views

Why is $\sum\left(\left\lfloor\frac{x}{p}\right\rfloor+\left\lfloor\frac{x}{p^2}\right\rfloor+\dots\right)\log p=\sum\frac{x}{p}\log p+O(x)$?

Why is $\sum\limits_{\substack{p:\text{prime}\\p\le x\\}}\left(\left\lfloor\frac{x}{p}\right\rfloor+\left\lfloor\frac{x}{p^2}\right\rfloor+\dots\right)\log ...
0
votes
0answers
35 views

Find all pairs of positive integers $(x,y)$ : $x(x+1) = y(y+1)(y+2)$

Find all pairs of positive integers $(x,y)$ : $$x(x+1) = y(y+1)(y+2)$$ I was able to find only two pairs: $(2,1)$ and $(14,5)$ and looks like no more exists. How to prove it?
0
votes
0answers
18 views

Reduced Residue class problem

I need to Prove that when $j \ge 3$, then every reduced residue class modulo 2j may be written in the form $((−1)^a)(5^b)$ , where a = 0 or 1 and $1 \le b \le 2^{j−2}$, and in which the integers a and ...
2
votes
1answer
90 views

Special representation of a number

How can I check, if a number $n$ can be representated by $$pq+rs$$ where $p,q,r,s$ are pairwise different prime numbers with the same number of digits. For example, $$105153899965560312960 = ...
0
votes
1answer
10 views

Let $g$ be a primitive root modulo $p^e$ for some $p$ prime, $e\geq 1$, show that gcd$(g,p)=1$

So far I've got: Suppose gcd$(p,g)\neq 1$, so $p\mid g$ and hence $p^e\mid g^e$ so $g^e\equiv 0 $ (mod $p^e$) Also $g^{p^{e-1}(p-1)}\equiv 1$ (mod $p^e)$ because $g$ is a primitive root. Not sure ...
2
votes
1answer
46 views

Consecutive numbers with less than $k$ prime factors?

Let $k$ be an integer. Consider the consecutive numbers with less than $k$ distinct prime factors. Are there arbitary large differences between those numbers ? With other words : Are there ...
0
votes
0answers
14 views

Difference between consecutive squarefree (cubefree) numbers

The jumping champions for the greatest difference between consecutive squarefree numbers are : ...
1
vote
1answer
32 views

Finding a rational point on $\mathscr{E} : y^2=x(x^2-25)$ to show $ \text{rank}(\mathscr{E})=1$

I'm trying to show that the rank of the following elliptic curve $$ \mathscr{E}: y^2=x(x^2-25)$$ is 1. Since it has a rational 2-torsion point at $(0,0)$, by considering the dual curve I've been ...
2
votes
1answer
50 views

Show that $a_n=\frac{1}{2}+\frac{1}{3}+…+\frac{1}{n}$ would not contain a natural number for all n [duplicate]

Show that the series: $a_n=\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}$ would not contain natural number for all n Can I prove that using "simple tools"?
0
votes
0answers
15 views

A Greatest Common Divisor Question

What is $GCD(a_0a_1\bmod N,a_0a_2\bmod N)$ where $GCD(a_0,a_1)$, $GCD(a_0,a_2)$, $GCD(a_1,a_2)$, $GCD(a_0,a_1,a_2)$ could each be non-trivial? ($a\bmod N$ here is remainder of $a$ divided by $N$).
2
votes
1answer
45 views

Suppose $m \mid 2^p - 1$. Show that $m \equiv 1 \pmod {2p}$.

I would like to get help with this proof: Let $p\ge3$ be a prime number, and let $m$ be a divisor of $2^{p}-1$, Prove that $m\equiv 1\ (mod\ 2p)$. I thought about proving that $m=1\ mod\ p$, ...
12
votes
5answers
175 views

Is 'Algebraic Number Theory' the study of the theory of algebraic numbers, or is it the study of the theory of numbers from an algebraic viewpoint?

Is Algebraic Number Theory the study of the theory of algebraic numbers? Or is it the study of the theory of numbers from an algebraic viewpoint? Or is it both? I know I can just find a wiki article ...
2
votes
2answers
48 views

Prove that there are infinitely many composite numbers n so that…

Prove that there are infinitely many composite numbers $n$ so that $n$ divides $3^{n-1}-2^{n-1}$. I proved $n=p^t$, where $p$ is a prime number and $t>1$, never satesfies the condition above.
1
vote
2answers
27 views

Simple Congruence Problem

-1 is a square modulo an odd prime if and only if that prime is congruent to 1 mod 4. Why is this, I cant seem to figure it out.
0
votes
0answers
30 views

Factorization of the sine

I am working on the Basel problem for a project for my Mathematics study. I need to proof that one could write the sine as a factorization of its linear roots. I know the proofs is in general done bye ...
6
votes
0answers
49 views

Meromorphic functions on $Y^2 = X^3 + 1$, genus.

Let $k$ be a field of characteristic $\neq 2$, and consider the quadratic extension $F$ of $k(X)$ generated by $\sqrt{X^3 + 1}$. What is/how do I find the genus of $F$? The progress I have so far: ...
2
votes
2answers
23 views

Describe all odd primes p for which 7 is a quadratic residue

I need to describe all odd primes $p$ for which $7$ is a quadratic residue. Now let $\left(\frac{a}{b}\right)$ be the Legendre Symbol. Then if $7$ is a quadratic residue $p$ we must have: ...
0
votes
1answer
49 views

Proof of No Unique Factorisation in $\mathbb Z[\sqrt{d}]$ for $d \leq-3$

How would I prove there is no unique factorisation in $\mathbb Z[\sqrt{d}]$ for $d \leq-3$, where $d$ is a square-free integer? I think it's something to do with the only invertible elements ...
6
votes
2answers
77 views

When is the sum of divisors a perfect square?

For $n=3$, $\sigma(n)=4$, a perfect square. Calculating further was not yielding positive results. I was wondering is there a way to find all such an $n$, like some algorithm? We know that if ...
0
votes
1answer
23 views

What are the quadratic residues of an odd prime? [duplicate]

I need to prove that -1 is a quadratic residue of an odd prime p iff p = 1 (mod 4) Any Ideas? Thanks
7
votes
1answer
98 views

The Diophantine Equation $x^2+y^4=2z^4$

We know that the Diophantine equation $x^2+y^4=2z^4$ has infinitely many solutions . Some of them are shown below $$(y,z)=(1,1),(1,13),(1343,1525),(2372159,2165017).$$ I investigated the ratio of ...
2
votes
3answers
30 views

Simple mod 7 problem

I need to Show that $7x^3 + 2 = y^3$ has no solutions in integers x and y. The solution I am given is: Suppose there are solutions to this equation. Then mod 7 we have $2 ≡ y^3$ (mod $7$) and hence ...
2
votes
0answers
26 views

Determination of quartic Gauss sums

Typically, the Gauss sum over $\mathbb{F}_p$ of order $k$ means the quantity $$\sum_{n=0}^{p-1} e^{2\pi i n^k/p}.$$ In the book Gauss and Jacobi Sums of Berndt, Evans and Williams, a more general sum ...
1
vote
1answer
33 views

Solving the congruence $3x^2 + 6x + 1 \equiv 0 \pmod {19}$

I tried to solve this equation but without a success: $3x^{2}+6x+1 \equiv 0 \pmod {19}$ I concluded hat $x(x+2)\equiv 6 \pmod{19}$, the only way i think to solve this is by just trying all the ...
5
votes
1answer
80 views

$1989|n^{n^{n^{n}}} - n^{n^{n}}$ for integer $n \ge 3$

Before anyone comments, yes this is kind of a duplicate of Prove that $1989|n^{n^{n^{n}}} - n^{n^{n}}$ . The problem that I'm having I don't see the $n=5$ as a counterexample. Also if anyone wants to ...
-1
votes
0answers
22 views

Divisibility by 9 with negative numbers [duplicate]

I know that the rule to check divisibility by 9 is to check if the sum of the digits of the number are divisible by 9. But what if the number is negative? Thanks in advance!
1
vote
4answers
56 views

Prove or disprove $\frac{\left(2^{p}-2\right)}{p}\ \in \Bbb N, \forall\, p,\, prime$

Apologies in advance for poor formatting, not completely accustomed to typeset. What I ask is any non-particular value p, with one condition that it is prime, for which to disprove the following ...
-1
votes
2answers
64 views

polynomials root finding [closed]

Is every root of a polynomial of positive integer degree n, and with a rational coefficients is considered algebraic number? and how one can find some roots to this polynomial ...
5
votes
2answers
66 views

Integral solutions to $56u^2 + 12 u + 1 = w^3.$

I would like to find all integer solutions to $$56u^2 + 12 u + 1 = w^3.$$ My computer thinks the only integral point is $(0,1).$ This problem arises from Integer solutions of $x^3 = 7y^3 + 6 y^2+2 ...
0
votes
0answers
15 views

The asymptotic upper density of $\{xy \colon 1\le x\le y\le 2x\}$

Find the asymptotic upper density of the set $\{xy\, \colon\, 1\le x\le y\le 2x\}$. In other words, let $S$ be the set of all integers which can be expressed as $xy$, for some positive integers ...
2
votes
1answer
15 views

Jacobi symbol problem

Let $n>3\newcommand{\jacobi}[2]{\genfrac{(}{)}{}{}{#1}{#2}}$ be an odd number. Find the value of the Jacobi symbol $\jacobi{n^3}{n-2}$. I know that $$\jacobi{n^3}{n-2}=\jacobi{n}{n-2}= ...
-3
votes
2answers
71 views

Can someone provide me a simplest way to calculate: [closed]

$42^{17} \pmod{3233}$ I know the answer is 2557 - But I need to know how to calculate this without help of a machine that generates the answer. Thank you!
1
vote
0answers
69 views

Solving $|x|^2=\sqrt{2}-1$ in $\mathbb{Z}[\xi_8] $

Is there a solution of the equation $|x|^2=\sqrt{2}-1$ in $\mathbb{Z}[\xi_8]$, where | | means the complex absolute value? In general, can I solve the equations of the form $|x|^2=c$ in each ring of ...
5
votes
3answers
65 views

Number of solutions of $x^2 + y^2 + z^2 = 0$ over finite fields.

I want to prove that the number of elements of the set $\{(x,y,z)\in \mathbb{F}_p^3: x^2 + y^2 + z^2 = 0\}$ is $p^2$. I know that the number of elements of the set is a multiple of $p$ using the ...
-6
votes
3answers
115 views

Is this the real reason why 1 is not prime? [duplicate]

Divisibility by 1 is misleading as it does not divide a number into smaller parts. If divisibility by 1 is disallowed, then: The Unit: A whole number that is indivisible. Prime: A whole number that ...
3
votes
1answer
48 views

Is this reasoning correct for average prime gap?

Since \begin{align} &\operatorname{li}(n)\sim\Pi (n)\equiv\sum _{k=1}^{\lfloor \log (n)\rfloor } \frac{\pi \left(n^{1/k}\right)}{k}\\ \end{align} then the average gap for \begin{align} ...
0
votes
0answers
44 views

Solve an equation of the prime counting function

The problem is, Find all the positive integral values of $x$ for which we have, $$\pi(p_n-x)=\pi(p_{n+1}-x-1)$$where $\pi(x)$ denotes the number of primes not exceeding $x$. I don't know where ...
1
vote
1answer
45 views

Number Theory/Quadratic Number Rings

Show that if $u,v,x,y$ are positive integers for which $u^2+2v^2=x^2+2y^2=p$ a prime number, then $u=x$ and $v=y$. I get that if we had $\alpha=u+v\sqrt{-2} \in \mathbb{Z}[\sqrt{-2}]$, then the ...
8
votes
0answers
64 views

Is there any known application for normal numbers?

Background: I am writing a master thesis on the complexity of the expansions of algebraic numbers in some complex basis $\beta$ with $|\beta| > 1$. This is a very small step towards proving the ...
0
votes
1answer
47 views

Order of element in algebraic group

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...
0
votes
2answers
17 views

Existence of a solution to a congruence

I need to check if congruence $$x^2+8x+69\equiv 0\pmod{271},$$ has a solution. How should I approach this? Checking all $271$ possible solutions, is obviously not intended.