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.

learn more… | top users | synonyms (1)

-2
votes
1answer
39 views

How many Gaussian Integers $z$ divide 10 [on hold]

How many Gaussian Integers $z$ divide 10, in that $10=z\times{w}$ for some Gaussian Integer $w$?
1
vote
0answers
29 views

Solvability of the congruence $(x+a)^n\equiv x^n\pmod p$ in $x$

When thinking about solving the diophantine $x^n-y^n=1001$ I noticed that my knowledge about the prime factorisation of $x^n-y^n$ does not suffice to attack such diophantines in general. Note I'm ...
1
vote
0answers
24 views

Number of squarefree numbers and the Basel problem

Who discovered/proved that there are about $$ \frac{x}{\zeta(2)} $$ squarefree numbers up to $x$, or (roughly) when was this first known? Today I think this is considered 'obvious', but I don't know ...
1
vote
0answers
10 views

A Mertens-like product over primes

MathWorld's page Prime Products gives the 'related result' (7) to Mertens' theorem: $$ \lim_{n\to\infty}\log p_n\prod_{k=1}^n\frac{1}{1+1/p_k}=\frac{\pi^2}{6e^\gamma}. $$ Does this identity have a ...
0
votes
1answer
47 views

How is Round(11) equal to 3?

I saw this on a Mathematical clock face. $1= \tan(45^{\circ})$, $2= \sqrt{4}$, $3=Round(11)$, and so on. How does $Round(11)$ equal $3$? I was told it has to do with unicode but I could not find it ...
4
votes
3answers
39 views

Find all 4 digits numbers that $ABCD=(CD)^2$

Please help me to solve following problem: Find all 4 digits numbers such that $ABCD=(CD)^2$.(any of $A,B,C,D$ is a digit!) I know one of solutions is $5776=(76)^2$.
0
votes
2answers
45 views

Does $O(\log^2(x))$ imply $O(x)$

Does $O(\log^2(x))$ imply $O(x)$ I have to prove the following: $$\sum\limits_{\substack{n\in\mathbb N\\n\le x}}\Lambda(n)\log(n)=\psi(x)\log(x)+O(x)$$ Applying partial sum I get; ...
0
votes
2answers
31 views

How to find kth smallest value of a linear equation

Here's a question that was asked in IOITC 2009 India. Even though it should have a solution related to algorithms, yet I post it here as it is pretty "number-theoretic". Indraneel loves posing ...
0
votes
3answers
53 views

How write a periodic number as a fraction? [duplicate]

What I call as a periodic number is for exemple $$0.\underbrace{13}_{period}131313...$$ or $$42.\underbrace{465768}_{period}465768465768.$$ So how can we put theses numbers like a integer ...
0
votes
1answer
29 views

Show that the set $\mathbb{Q}^+$ is a group under ordinary multiplication

To be a group, a set with a binary operation has to satisfy all four of the group axioms. My problem is with closure as each time I am unsure if my proof suffices. The set of positive rational numbers ...
0
votes
1answer
13 views

Show the following including $\sigma$ function

How do I show that $\sigma (2k)=4k$ if and only if $k=2^{p-2}(2^p-1)$ where $2^p-1$ is a prime number. I want to show that if $k$ is odd and $\sigma(k) = 2k$ then $k=p^am^2$ for some p with $(p,m)=1$ ...
2
votes
2answers
48 views

Prove that $a^{p^2}\equiv a^p\pmod{p^2}$ for every $a\in\mathbb{Z}$

Question: Prove that $a^{p^2}\equiv a^p\pmod{p^2}$ for every $a\in\mathbb{Z}$ First note that when $p$ is prime and $1\leq r\leq p-1$ the binomial coefficient $$\binom{p}{r}=\frac{p!}{r!(p-r)!}$$ is ...
2
votes
0answers
49 views

can you help me to solve this equation in antural numbers set?

Can you help me find the natural solutions of $$2^x+3^y=5^z$$ or can you introduce a book that talk about these equations?
1
vote
0answers
29 views

A question about a property of Gauss sum.

I am reading the book and I have some questions about Gauss sum. The Gauss sum is defined in the end of page 4, formula (1.14), by \begin{align} g(m,c)=\sum_{a \mod c} \left( \frac{a}{c} \right)_n ...
0
votes
2answers
41 views

Confusion with $O$ function

I read this identity in lecture notes and need help understand ing the $O$ function $$\sum_{1\leq d\leq x}\mu(d)\cdot \frac{1}{2}\left\lfloor\frac xd\right\rfloor\left(\left\lfloor\frac ...
2
votes
0answers
25 views

Lower and upper density of iterations of subsets of $\mathbb{N}$

For $A\subseteq \mathbb{N}$ we define the lower and upper density by if $$\text{lowd}(A)=\text{lim inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}, \text{upd}(A)=\text{lim ...
3
votes
0answers
23 views

The copy-problem : Does any block of digits appear at least twice?

Suppose, $N$ random digits have been generated. Let $X$ be the largest natural number with the following property : There are natural numbers $i$ and $j$ with $i+X-1<j$ , such that the digits $i$ ...
6
votes
1answer
124 views

Does this curiousity have any meaning?

If $\pi$ is calculated to the $360$-th digit after the decimal point, the last digits are $360$ : ...
1
vote
1answer
30 views

Testing randomness

I'm looking for informations about randomness and especially - random numbers. I found some about random number generators, but for now, the question, that concerns me is how statistically differ ...
0
votes
1answer
30 views

Application of Gauss' lemma

Using Gauss' lemma show when $p$ is an odd prime, one has $$\left(\frac{2}{p}\right)=(-1)^{(p^2-1)/8}$$ The proof starts with let $a=2$ in gauss' lemma, then one has $$ a_j = \begin{cases} 2j, ...
3
votes
3answers
684 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
31 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
47 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
13 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 ...
2
votes
0answers
53 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 $ ā„ $ 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) := ...
3
votes
0answers
42 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
43 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
32 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 $((āˆ’1)^a)(5^b)$ , where a = 0 or 1 and $1 \le b \le 2^{jāˆ’2}$, and in which the integers a and ...
2
votes
1answer
84 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 ...
1
vote
1answer
24 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
31 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
49 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 $m=1\ mod\ p$, ...
10
votes
4answers
163 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
24 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
29 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
47 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
48 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
75 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
93 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 ...