Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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$r$ primitive root of prime $p$, where $p \equiv 1 \mod 4$: prove $-r$ is also a primitive root

question is as follows: Let $p$ be a prime with $p \equiv 1 \mod 4$, and $r$ be a primitive root of $p$. Prove that $-r$ is also a primitive root of $p$. I have shown that $-r^{\phi(p)} \equiv 1 ...
2
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1answer
24 views

Numerical verification of the ternary Goldbach conjecture

In his proof of the ternary Goldbach conjecture, H.A. Helfgott says that it has been verified that every odd number less than $N_0 = 10^{30}$ is the sum of at most 3 primes. How would one verify this ...
2
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1answer
36 views

Problem on Number of Quadratic Residues

We have two primes $p,q$ and an integer $a$ such that $$\gcd(a,pq)=1$$ How to prove that for the following congruence $$x^2 \equiv a \mod pq$$ either there will be $4$ solutions or $zero$ solutions. ...
3
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2answers
84 views

elementary proof that infinite primes quadratic residue modulo $p$

$p \gt 2$ is a prime, then there are infinite primes $q$ such that $q$ is a quadratic residue modulo $p$. With Dirichlet's theorem on arithmetic progressions, the problem is easy! How about ...
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4answers
77 views

Number theory with positive integer $n$ question

If $n$ is a positive integer, what is the smallest value of $n$ such that $$(n+20)+(n+21)+(n+22)+ ... + (n+100)$$ is a perfect square? I don't even now how to start answering this question.
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3answers
48 views

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 ...
2
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1answer
51 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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1answer
99 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 ...
1
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1answer
54 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
67 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 ...
2
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1answer
45 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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0answers
23 views
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0answers
27 views

Quotient is a square [duplicate]

Find all primes $p$ for which the quotient $\frac{2^{p-1}-1}p$ is a perfect square. I know a solution $p=7$ but I couldn't prove. And I don't know this is only solution or not.
2
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1answer
52 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 $\mathbb{C}$ ...
2
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2answers
35 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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0answers
24 views

Find point of maximal order $\operatorname{lcm}(a,b)$ on a curve

In Elliptic curve okamoto uchiyama there is a condition for picking base point $G$ such that $G$ belongs to $E_n$ of maximal order $\operatorname{lcm}(|E_{p2}|,|E_q|)$. I am not getting these ...
2
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2answers
58 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!
2
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1answer
33 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 ...
2
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2answers
71 views

p odd prime. Prove that if $a\equiv b~(mod~p)$ then $a^p\equiv b^p~(mod~p^2)$. Hence 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 $a\equiv b~(mod~p)$ then $a^p\equiv b^p~(mod~p^2)$. Hence show the Diophantine equation $x^5+y^5=z^5$ has no integer solutions with $5\not\mid xyz$. ...
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1answer
37 views

Bijection and image

Let f: A -> B be a bijection, so f^-1: B -> A is a function. Let X be a subset of A. How do I prove that Im(f)(X) = Preim(f^-1)(X)? Thank you.
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1answer
39 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
24 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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2answers
36 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
62 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 ...
3
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3answers
70 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
42 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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2answers
499 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
34 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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0answers
12 views

Diophantine equation with 6 variables.

In this equation: $aX^2+bY^2+cZ^2=abc+2XYZ+F$ $F$ - integer number given by the condition of the problem. A rather Tran decision: $a=(2pk-p^2+p-k^2)((t-s)^2-1)+2tsk+p(1-t^2)-(2k-p+1)s^2+F$ ...
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1answer
52 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
15 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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22 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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5answers
106 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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46 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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4answers
60 views

solve and explain the Diophantine equation [closed]

Solve Diophantine equation and find the value of $x$ and $y$. For the value of $x$ and $y$ we solve through Diophantine equation. $$199x -98y = -5 $$
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2answers
46 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
73 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 this ...
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0answers
29 views

Application of Fermat's little theorem [duplicate]

show that $a^{13}\equiv a \pmod {35}$ using Fermat's little theorem. Use Fermat's little theorem with primes $5$ and $7$.
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1answer
42 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
40 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
39 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
21 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
28 views

Question on perfect numbers: $\sigma (n) = n \rightarrow \sigma (kn) = kn$? [duplicate]

I'd like to proof that if $\sigma (n) = n$, where $\sigma(n)$ is the sum of all divisors of n$ then follows $\sigma (kn) = kn, k \not= 1$. I have $\sigma (kn) = \sum_{x | kn} x = ...??$
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0answers
27 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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2answers
51 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
17 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 ...
0
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0answers
45 views

number theory expressing as notation [closed]

Surf the internet and find a theorem of number theory. State the claim of the theorem, and then express it in logical notation. (Example: It is a theorem of number theory that if n is an even natural ...
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0answers
57 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
23 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. ...