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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1answer
19 views

Is $(a^s\pmod{n})^k = a^{sk}\pmod{n}$?

Is $(a^s\pmod{n})^k = a^{sk}\pmod{n}$? And if it is, how come? I've thought about it and this is the only thing that makes sense (for now).
17
votes
2answers
269 views
+50

Without using Heegner-Stark-Baker, $\mathbb{Q}(\sqrt{-11})$ has class number $1$.

Prove that $\mathbb{Q}(\sqrt{-11})$ is of class number $1$. I have found that the ideal $(2)$ of the integer ring $\mathbb{Z}[(1 + \sqrt{-11})/2]$ of $\mathbb{Q}(\sqrt{-11})$ is a prime ideal. ...
0
votes
0answers
9 views

Number of number in range $(l, r)$ satisfying XOR constarint

Here's a questions that's been bugging for some time now: Define the set $S_n = \{k \oplus (k + n)\mid k \in \Bbb Z, k ≥ 0\}$ (here, $\oplus$ is bitwise exclusive OR). To put it another way, $x$ ...
0
votes
0answers
22 views

Elliptic curves and the reduction map

For $n \geq 1$ we write $E_n = \{(x:y:z) \in \ker(\pi)|x/y \in p^n \mathbb{Z}_p\}$ with $\pi: E(\mathbb{Q}_p) \rightarrow E(\mathbb{F}_p)$ the reduction map. I know that $\pi$ is a group morphism and ...
2
votes
1answer
26 views

$a,b,N$ are integers. Prove $x=x_0+\cdots$, $\ \ y=y_0+\cdots $ are solutions to $ax+by=N$

I'm asked to prove that if $a,b,N$ are integers, then in the equation: $$ax+by=N$$ I must prove that the integers $$x=x_0+\frac{b}{d}t,\ y=y_0-\frac{a}{d}t$$ are solutions to the equation. where ...
15
votes
8answers
827 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?
1
vote
1answer
30 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 ...
0
votes
0answers
21 views

Set theory, intersection of two sets

We have the set $D$ which consists of $x$, where $x$ is a prime number. We also have the set F, which consists of $x$, belongs to the natural numbers (positive numbers $1, 2, 3, 4, 5,\dots$) that is ...
-1
votes
1answer
35 views
0
votes
0answers
21 views

Least pair of numbers having at least $k$ distinct prime factors

Consecutive numbers with less than $k$ prime factors? shows that for every $k$, there is a pair $(n/n+1)$, such that $n$ and $n+1$ both have at least $k$ distinct prime factors. The object is to ...
0
votes
2answers
34 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 ...
3
votes
2answers
61 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 ...
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 ...
4
votes
2answers
37 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 ...
5
votes
2answers
60 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 ...
5
votes
1answer
38 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
2answers
44 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 ...
0
votes
1answer
23 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, ...
4
votes
4answers
124 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 ...
2
votes
1answer
18 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 ...
1
vote
0answers
35 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 : ...
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 ...
2
votes
1answer
42 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 $\mathfrak p$ ...
0
votes
1answer
36 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) ...
2
votes
0answers
19 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 = ...
0
votes
1answer
24 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 ...
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. ...
-3
votes
0answers
266 views

Fermat's last theorem generalization [closed]

Conjecture: Let $g$ is a positive algebraic number greater than two, then the equation ($x^g+y^g=z^g$) doesn't have any solution, where ($x, y$ and $z$) are three distinct positive coprime integers ...
4
votes
1answer
49 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 ...
0
votes
0answers
43 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 ...
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
1answer
230 views

Increasing sequence of divisors of a quadratic trinomial

This question is from a Russian contest, the 2011 Tuymaada Olympiad. It's the fourth question on day two for the problems at grade level 2. Let $P(n)$ be a quadratic trinomial with integer ...
-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
66 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 ...
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
0answers
88 views

Blocks of consecutive natural numbers

Let a < b be natural numbers. Prove that every block of b consecutive natural numbers contains two distinct elements whose product is divisible by ab. Suppose now a < b < c are natural ...
5
votes
1answer
176 views

On odd perfect numbers $N$ given in the Eulerian form $N = {q^k}{n^2}$

Note: This question was cross-posted from MO. Preamble: I apologize in advance if this particular MSE post would appear to be a bit of a polymath approach, I just had to put down all the details to ...
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 ...
0
votes
1answer
306 views

How to determine the key-matrix of a Hill cipher where the encrypted-message-matrix is not invertible?

I am new to this subject and I have a homework problem based on Hill cipher, where encryption is done on di-graphs (a pair of alphabets and not on individuals). The alphabet domain is $\{A\dots ...
-2
votes
1answer
49 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 $w$?
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 ...
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 ...
0
votes
2answers
52 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 ...
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
3answers
56 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
109 views

Gaps between primes: bounds - a question of possibilty

Let $n$ be any given natural number. Let $p$ be the very next prime greater than $n$. Let $b$ be the bound for the prime gap above $n$. Here, the bound is strictly the limit from $n$ to $p$, meaning ...
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 ...
10
votes
1answer
228 views
+50

Numbers having in decimal representation no common digits with all their proper divisors

Let us call a positive integer having in decimal representation no common digits with all its proper divisors "a good number". $54$ is a good number : $1,2,3,6,18,27$ $48$ is not a good number : ...