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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0
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0answers
8 views

Show the following including number of divisors d(n)

I know how to show that $(d ∗ \mu)(n) = 1$ for all n ≥ 1.But.. I have two solutions. Firstly... result is trivial, because $d = 1 ∗ 1$ Secondly We know that both sides are multiplicative. Thus it ...
2
votes
1answer
27 views

My proof that an n digit number, times an n digit number can be expressed as a 2n digit number

I am very proud to say this is the first time I've actually used maths to endeavour to prove something without it being related to a question from my course! Statement In a base $B$, an $n$ digit ...
4
votes
0answers
19 views

Remainders of quadratic trinomial

The problem is to determine, whether there exist a quadratic trinomial $f(x) = ax^2 + bx +c$ with integer coefficients (with $a$ not a multiple of 2014), such that the numbers $\ f(1), \ f(2),\, ...
0
votes
3answers
30 views

Given a fixed positive integer $k$, find the number of pairs of integers $(x,y) \in \mathbb{Z}^2$ such that $x^2+y^2=5^k$

Given a fixed positive integer $k$, find the number of pairs of integers $(x,y) \in \mathbb{Z}^2$ such that $$x^2+y^2=5^k$$ Attempt: Clearly $x$ and $y$ cannot have the same parity. Assume that ...
5
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1answer
29 views

Find all solutions in N of the following Diophantine equation

$(x^2 − y^2)z − y^3 = 0$ i divide by $z^3$ and look for rational solutions of the equation $A^2 − B^2 − B^3 = 0.$ The point $(A,B) = (0, 0)$ is a singular point, that is any line through this point ...
-1
votes
0answers
32 views

Number theory / decimal representation

Prove that for any $n\in\mathbb{N}$ there exists a number $m\in\mathbb{N}$ such that the decimal representation of $m^2$ has $n$ ones at the beginning and some combination of $n$ ones and twos at ...
-1
votes
3answers
58 views

Confused about transcendental numbers [on hold]

I'm little confused about the type of numbers that had been known, for example, consider a polynomial equation with rational and irrational coefficients of a degree p-prime number that is greater than ...
3
votes
2answers
56 views

Prove or disprove that $a^{\phi(n) + k} \equiv a^{k} \mod{n}$

Prove or disprove that $$ a^{\phi(n) + k} \equiv a^{k} \mod{n} $$ where $\phi(n)$ is Euler's totient function, for all positive integers $a$ and $n$, as long as $k$ is $\geq$ the ...
-2
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1answer
47 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
1answer
48 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 ...
3
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0answers
33 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 ...
4
votes
1answer
23 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
49 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
40 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
48 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
32 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
50 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?
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0answers
31 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
42 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
129 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
31 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
689 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
32 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
49 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
14 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
56 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 ...
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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) := ...
3
votes
1answer
45 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
33 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
88 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
25 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
172 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.