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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Find all natural numbers $x$ such that …

Find all natural numbers $x$ such that product of their digits is equal to $x^2-10x-22$ I've already found one such number it is $12$ because $(12)^2-10\cdot12-22=2$ and $1\cdot2=2$ but I don't know ...
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1answer
145 views

Modular Form, but Cusp Form in Disguise

Suppose I have a holomorphic modular form $f \in M_k(\Gamma_0(N), \chi)$ with $k \in \mathbb Z^+$ and $f = \sum_{n=1}^\infty a(n)q^n$. Considering the $q$-expansion, one may suspicious that $f$ is ...
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2answers
160 views

Primality of the number of obtained by concatenating the n consecutive digits

Let $f_n$ be the number obtained by concatenating the first $n$ numbers (in base 10). For example $f_1 = 1, f_3 = 123$ and $f_{13} = 12345678910111213.$ Now if $n$ is even or divisible by $5$ then ...
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1answer
72 views

Is it true that $6n^2+p$ gives primes for $n=0,1,2,\dots,p-1$ iff $Q(\sqrt{-6p})$ has class number $4$?

Let $p$ be a prime number, are the following statements true? 1.Quadratics of the form $6n^2+p$ gives primes for $n=0,1,2,\dots,p-1$ iff $Q(\sqrt{-6p})$ has class number $4$. And all such primes ...
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2answers
97 views

Proof that 2 and 3 are the only siamese twins that exist!

Let us say that two prime number p and q are siamese twins if |p-q|=1. List all the siamese twins that exist, and prove your list is complete. Proof: 2 and 3 are prime numbers and 3-2=1. Therefore 2 ...
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1answer
53 views

What does notation like $A^1_0$ and $f^2_1$ mean?

My logic book uses these symbols to represent predicates and constants, respectively (with varying numbers in the bottom/top), but I don't really know what they mean. Any help? From the book: ...
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248 views
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1answer
78 views

Existence of finite morphism to projective line

As we all know, Belyi's theorem says: A complex curve $X$ is defined over a number field, if and only if there exists a finite morphism $t:X\to \mathbb{P}^1_\mathbb{C}$ of varieties over $\mathbb{C}$ ...
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2answers
69 views

Prove that if b is coprime to 6 then $b^2 \equiv 1 $ (mod 24)

Let $\gcd(b,6) = 1$. Prove that $b^2 \equiv 1 $ (mod 24). Now I have that as $\gcd(b,6) = 1$, we know that $3\nmid b $ and $2\nmid b$ (else the GCD would be 3 or 2 resp.) So as $2\nmid b$, $b$ must ...
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2answers
79 views

A ring automorphism in cyclotomic field

My friend asked me a question, see this. I've thought about that for some time, but I cannot do it, I don't want to let her wait too long, can you explain it for me? Thanks in advance!
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1answer
59 views

distance between irreducible elements in a number ring

Consider the number ring $\mathbb{Z}[\phi]$ where $\phi$ is the positive root of $X^2-X-1$. Any of its elements can be written as $a+b\phi$ with $a$ and $b$ integers. There is a norm $N$ such that ...
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2answers
153 views

p odd prime. Prove that if $a\equiv b\pmod p$ then $a^p\equiv b^p\pmod p^2$. Then show $x^5+y^5=z^5$ has no integer solutions with $5\not\mid xyz$

Question: Let $p$ be an odd prime. Prove that if $a\equiv b \pmod p$ then $a^p\equiv b^p \pmod p^2$. Then show the Diophantine equation $x^5+y^5=z^5$ has no integer solutions with $5\not\mid xyz$. My ...
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71 views

A (possibly) easier version of Bertrand's Postulate

While attending a math puzzle contest, my friend (a math student) asked me to prove that $$\sum_{k=1}^n \frac{1}{k} \notin \mathbb{Z} \quad \forall n \geq 2$$ Being the first time seeing this ...
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2answers
34 views

Are there arbitrarily large arithmetic progressions of primes for some fixed progression width?

Is there any positive integer $b$ so that for any positive integer $k$, there exists positive integer $a$ so that all $a + bn$ are prime for all $1 \leq n \leq k$? My guess is not, what's a proof?
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41 views

Different values of $x$ and $y$ between $\sqrt{39}$ and $\sqrt{224}$

If $x$ and $y$ are whole numbers between $\sqrt{39}$ and $\sqrt{224}$, then how many different values can $x$ + $y$ have? OK, first I found that the set numbers are: $$7, 8 ,9 ,10 ,11 ,12, 13,14$$ ...
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2answers
98 views

What does $conclude$ mean in this sentence?

My friend asked me a question, but I don't know the meaning of the sentence Conclude that $\sigma_n$ is a ring automorphism here, does it mean Prove that $\sigma_n$ is a ring automorphism or Make the ...
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3answers
94 views

Prove that integers relatively prime.

I need help to prove that $$\gcd(n, 2^{2^n} + 1)=1,\ n = 1,2,\dots$$ I have no idea how start the proof.
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1answer
55 views

How find this positive integer $(a,b)$ such $\left(\frac{a}{b}-\left[\frac{a}{b}\right]\right)\left[\frac{a}{b}\right]=2013$

let $a,b$ is positive integer numbers,and such $$\begin{cases} \gcd(a,b)=1\\ b\le 100\\ \left(\dfrac{a}{b}-\left[\dfrac{a}{b}\right]\right)\left[\dfrac{a}{b}\right]=2013 \end{cases}$$ Find the pairs ...
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Let $p$ be a prime, and let $q$ be a prime factor of the Mersenne number $2^p-1$. Prove that $ord_q(2)=p$ and deduce that $q\equiv 1 \;(mod\; p)$

$q\mid (2^p-1)$ $\Longrightarrow 2^p-1\equiv0 \;(mod\; q)$ $\Longleftrightarrow 2^p\equiv 1 \; (mod \; q) $ so $ord_q(2)\leq p$ but $2^p\equiv 2^0 \; (mod \; q) \Longleftrightarrow p\equiv 0 \; ...
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667 views

Riemann's zeta as a continued fraction over prime numbers.

Riemann's zeta function is a function with many faces, I mean representations. I recently derived this one, bellow, as a continued fraction over prime numbers. $$ \zeta(s)=1 ...
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1answer
58 views

Evaluate smartly a function on a multiplication grid

I am asking myself the following question: Suppose one has a grid $G \in \mathbb{N}^{n\times n}$ where $g_{ij} = i\cdot j$, $i,j \leq n$. I would like to evaluate a function $f: G \to \mathbb{N}$. ...
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1answer
60 views

Difference between sum of first n primes and prime(prime(n))

The seq is: -1, 0, -1, 0, -3, 0, -1, 10, 17, 20, 33, 40, 59, 90, 117, 140, 163, 218, 237, ... http://oeis.org/A239731 Is there's a formula looks like $$a(n) =n^2logn/2$$ for this seq?
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2answers
29 views

prove that $\forall k\in \mathbb Z$ if $(m,n)=1$(coprimes) then $(k,m,n)=1$

Prove that $\forall k\in \mathbb Z$ if $(m,n)=1$(coprimes) then $(k,m,n)=1$ I did this by contradiction: let $d=(k,m,n)$ so that $d\neq 1$; by definition of greatest common divisor $d|k, d|m, d|n$ ...
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0answers
81 views

The smallest class of numbers closed under addition, multiplication, and exponentiation

Let $\def\A{\mathfrak A}\A$ be the smallest subset of $\Bbb C$ that contains the algebraic numbers and also all numbers of the form $$\sum \alpha_i^{\beta_i}$$ where the $\alpha_i, \beta_i$ are ...
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4answers
129 views

Is this a true theorem? [closed]

I'm trying to prove the existence of the following theorem: If $n,p \in \mathbb{N}$, then $(p+1)^n = 1 \mod p$ Is this theorem true? I think it is, but I don't know how to prove it! Thanks!
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1answer
97 views

Integer Points on Circles

Let $r(n)$ denote the number of integral solutions to $a^2+b^2 = n$ where $a,b,n$ are integers. Furthermore, we count the pairs with regard to order and signs. (So if $(a,b)$ is a solution, so are ...
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142 views

Proving divisibility tests using congruence relations [closed]

For a positive integer $N$ which has the decimal representation $$N=\sum_{k=0}^n a_k\cdot10^k $$ Prove that $$11\mid N \Longleftrightarrow 11\mid \sum_{k=0}^n(-1)^k a_k $$ using congruence ...
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3answers
6k views

First 10-digit prime in consecutive digits of e

Problem. What is the first $10$-digit prime in consecutive digits of $e$. For those of you who don't know, in 2004 the answer produced a URL to a Google employment page (sort of). I just found about ...
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1answer
53 views

question on combinatorics and number theory

We have an equation as: $a\times b < n$ where $n$ is any positive integer. Now my question is how many pairs of positive integers $(a,b)$ can be found to satisfy the equation. For example, ...
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4answers
57 views

How to find $k$ in $6^{k} \equiv 2 \mod {13}$

Find for which $k$ is $6^{k} \equiv 2 \mod {13}$ I'm having trouble with these types of question in my cryptography class. This is part of Diffie–Hellman algorithm for calculating a shared key. ...
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1answer
85 views

Euclid's first theorem/ Euclid's lemma

How to prove that if $c$ divides $ab$ and $\operatorname{gcd}(a,c)=1$, then show that $c$ divides $b$. that means if $c|ab$ and $(a,c)=1 \implies c|b$.
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1answer
26 views

Need help proving cardinal of $\{n \in \mathbb{N}: n \le x, d|n\}= \lfloor \frac{x}{d} \rfloor$

I need to show this $\{n \in \mathbb{N}: n \le x, d|n\} = \lfloor {\frac{x}{d}} \rfloor$ but I don't know where to start =(
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0answers
30 views

Finding the summation of the series. [duplicate]

Is there any formula to find out the summation of the series. $$\sum_{i=1}^{n} \lfloor \frac{n}{i} \rfloor$$ Can someone help me with this.
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3answers
687 views

Determine the last two digits of $3^{3^{100}}$

Determine the last two digits of $3^{3^{100}}$ This is one of the problems in the past exam my modern algebra course. I think I need to use euler-fermat theorem but can't figure out how to use it for ...
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1answer
85 views

Solve this system of linear congruences: 7x+3y is conruent to 10(mod 16) and 2x+5y is congruent to 9(mod 16).

Solve this system of linear congruences: $7x+3y$ is conruent to 10(mod 16) and $2x+5y$ is congruent to 9(mod 16). I've looked at similar questions and for some reason I can't get an answer to come ...
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0answers
165 views

Proof Synopsis of Fermat's Last Theorem

I'm taking a introduction to higher math course now (mostly number theory) and our professor wants us to include two sentence proof synopses with our longer proofs. This got me thinking, What is a ...
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1answer
115 views

Truncatable primes

Why only 11 ? The number 3797 has an interesting property. Being prime itself, it is possible to continuously remove digits from left to right, and remain prime at each stage: 3797, 797, 97, and 7. ...
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70 views

Proportional growing number set $\mathbb{X}\subset\mathbb{N}$

1. Question: Is there such a set of numbers $\mathbb{X}\subset\mathbb{N}$, in which the proportion of product and sum of all natural numbers $n\in \mathbb{N}$ grow proportional? $$\begin{equation*} ...
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1answer
32 views

is the lattice [$2+\sqrt{11},3-2\sqrt{11}$] an ideal in $O_{11}$

Is the lattice [$2+\sqrt{11},3-2\sqrt{11}$] an ideal in $O_{11}$ $N(2+\sqrt{11})=4-11=-7$ $N( 3-2\sqrt{11} )=9-4*11=-35$ $Tr( \left(2+\sqrt{11})(3-2\sqrt{11} )\right)=56$ since ...
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1answer
176 views

if $\frac{a_{1}a_{2}a_{3}\cdots a_{n}-1}{(a_{1}-1)(a_{2}-1)(a_{3}-1)\cdots(a_{n}-1)}\in N^{+}$,How find $a_{i}$

Question: let $a_{1},a_{2},\cdots,a_{n}$ is positive integer, and such $1<a_{1}<a_{2}<\cdots<a_{n}$ and $$\dfrac{a_{1}a_{2}a_{3}\cdots ...
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2answers
247 views

A different Harmonic series.

Let's call the following numbers than can be produced by playing with plus and minus: $$H_n'=\pm\frac{1}{1}\pm\frac{1}{2}\pm\frac{1}{3}\pm\cdots\pm\frac{1}{n}$$ "Harmonic kids" of $H_n$. We have a ...
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2answers
154 views

order of an integer related

I was reading a number theory text and this is when I encoutntered a line like this: "for $n=12$ , $\phi(12)=4$, yet there is no integer that is of order $4$ modulo $12$; indeed we find that ...
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117 views

What is the time to create a complement of a graph?

It seems to me that the running time to make a complement of a graph with $n$ nodes is $n!$. Is there any way to make this running time polynomial? That is, is there any method to construct a ...
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1answer
132 views

Derivation of a generalization of Mertens' Third Theorem.

One of Mertens' Theorems states $$\prod_{p\le x}(1-\frac{1}{p})\sim \frac{e^{-\gamma}}{\ln(x)}.$$ I have seen a generalized version that states $$\prod_{m<p\le x}(1-\frac{m}{p})\sim ...
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0answers
100 views

Coprimality and division

I'm trying to understand 100% intuitively and rigorously ( at the same time ) almost all facts in basic number theory. I'm going really slow-paced and at the moment i didn't reach primes and unique ...
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1answer
76 views

Sum of Euler Phi equalities

Show: $\sum_{n\le x} \phi(n) [\frac{x}{n}] = \sum_{n \le x} \sum_{m\le \frac{x}{n}} \phi(m)$ I know the left most sum boils down to $\sum_{n\le x} n$. If we know that $m|\frac{x}{n}$ then we know ...
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3answers
381 views

Distribution of prime numbers. Can one find all prime numbers?

I want to know if it is possible to find a formula that gives all the prime numbers? or can one find the distribution of prime numbers? I know that there is a set of ongoing research on prime ...
2
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2answers
58 views

Solve an equation in positive integers

Does $$x^2+y^2=3(z^2+ u^2)$$ have solutions in positive integers? I was assigned this problem, but I am struggling to find a solution. I guess that a proof by contradiction is required.
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35 views

to show one to one function

how we can show f(x)=-x^2+6x-7 , if x is less than or equal 3 one to one?? DO we use a quadratic formula?? f(x)=-x^2+6x-7= 2-(x-3)^2. f(x)=f(y) implies x=y which is injective or one to one. ...
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133 views

Given a congruence equation ax = b (mod n), how can I prove this GCD?

I am given the equation $ax = b (mod$ $n)$ and that $d = (a,n)$. Suppose that $x_o$ is a solution to the equation. I need to prove that d is the greatest common divisor of not only a and n, but b as ...