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
votes
1answer
21 views

Hexadecimal representation of tenths place of $\frac{A}{C}$

So we can write $$\frac{A}{C}= a_1 \times 10^{-1} + a_2 \times 10^{-2} + a_3 \times 10^{-3} +\cdots$$ How do I find the hexadecimal representation of $a_1$ where all numbers and variables are in the ...
0
votes
2answers
39 views

can this decimal number be converted into a fraction?

Can $$ 0.45647456647456664745666647456666647456666664745666666647456666666647\dots $$ be converted into a fraction of $\frac{N}{M}$ where $N$ and $M$ are integers? I know there is an algorithm ...
1
vote
0answers
16 views

Carmichael number proof

Prove that Carmichael number is squarefree. My solution: Assume that $n=p^kq$ is a Carmichael number and $k\geqslant 2$. Because gcd($(p+1)^{k-1},n)=1$, we get $$(p+1)^{p-1}\equiv 1 \pmod{p^{k}}$$ ...
0
votes
1answer
17 views

Find the number of possible values of $a$

Positive integers $a, b, c$, and $d$ satisfy $a > b > c > d, a + b + c + d = 2010$, and $a^2 − b^2 + c^2 − d^2 = 2010$. Find the number of possible values of $a.$ Obviously, factoring, ...
1
vote
0answers
29 views

Primes with the first $k$ digits of the solution of the equation $e^{-x^2}=x$

Let $s$ be the solution of the equation $e^{-x^2}=x$ The first $1000$ digits are : ...
2
votes
1answer
17 views

Sequences formed by integer evaluations of polynomials modulo $ p^{k} $, where $ p $ is a prime number and $ k \in \Bbb{N} $.

I have the following question. Let $ p $ be a prime number and $ k $ a positive integer. Let $ (a_{n})_{n \in \Bbb{Z}} $ be a two-way sequence in $ \Bbb{Z} / p^{k} \Bbb{Z} $. Then is it true that ...
2
votes
1answer
38 views

If $1+x+x^2+\cdots+x^{y-1}$ is a prime number then how prove that y is also a prime number?

If $1+x+x^2+\cdots+x^{y-1}$ is a prime number then how prove that $y$ is also a prime number? $x$ and $y$ are natural numbers
2
votes
0answers
13 views

Why is $m \infty$ the conductor of $K = \mathbb{Q}(\zeta_m)/\mathbb{Q}$?

Wouldn't this be saying that for all $p$ dividing $m$, $1 + p^{\operatorname{ord}_p(m)} \mathbb{Z}_p$ is contained in the group of local norms $N_{\mathfrak p/p}(K_{\mathfrak p})$, where $\mathfrak p$ ...
2
votes
0answers
14 views

Show that $U_i \rightarrow U_{i+e}, x \mapsto x^p$ is an isomorphism

Show that $U_i \rightarrow U_{i+e}, x \mapsto x^p$ is an isomorphism? Let $K$ be a finite extension of $\mathbb{Q}_p$ with uniformizer $\pi$, prime $\mathfrak p$, and ramification index $e = ...
4
votes
4answers
121 views

Show the following including triple statement

How do I show \begin{equation*} \sum \limits_{n=0}^{\infty} z^n=\prod \limits_{m=0}^{\infty}(1+ z^{2^m}) = (1-z)^{-1}? \end{equation*} The very left side is obvious because it is the geometric ...
0
votes
1answer
35 views

$f'(x) \equiv 0 \pmod{p}$ with $\deg f < p$ implies $f(x) \equiv c \pmod{p}$

Let $f(x) = P(x)/Q(x)$ where $P, Q \in \mathbb{Z}[x]$. Define $\deg f = \max(\deg P, \deg Q)$. Then as usual, $f'(x) = (Q(x)P'(x) - P(x)Q'(x))/Q(x)^2$. Suppose for some prime $p$, we had $f'(n) ...
1
vote
1answer
21 views

A Problem on the Prime Counting Function $\pi(x)$

Let $\pi(x)$ denotes the number of primes less than or equal to $x$. Also suppose that for some fixed $N$ we have $\pi(x+y)\ge\pi(x)+\pi(y)$. The problem is, Show that the equality in the above ...
13
votes
7answers
488 views

Computing irrational numbers

I am genuinely curious, how do people compute decimal digits of irrational numbers in general, and $\pi$ or nth roots of integers in particular? How do they reach arbitrary accuracy?
0
votes
0answers
19 views

Primality radius and quadratic reciprocity law

Given an integer $n>1$, I say that $r$ is a primality radius of $n$ if both $n-r$ and $n+r$ are primes. Goldbach's conjecture asserts that every integer greater than $1$ admits a primality radius. ...
0
votes
1answer
23 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 ...
3
votes
1answer
49 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 ...
5
votes
1answer
29 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
34 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
votes
2answers
54 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 ...
0
votes
0answers
42 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
63 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
64 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
votes
1answer
48 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
50 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
votes
0answers
35 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
26 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
50 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
42 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
50 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
33 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
55 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
14 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
51 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
32 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
26 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
132 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
32 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
692 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
50 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 ...
3
votes
0answers
60 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) := ...