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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-4
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0answers
31 views

find a method for twin primes and with Golbach conjecture [on hold]

There are infinitely many twin primes. Two primes (p, q) are called twin primes if their difference is 2. Let be the number of primes p such that p<= x and p + 2 is also a prime. a sample ...
0
votes
1answer
34 views

Find the missing digit in the number 23104*791

Find the missing digit in the number $23104*791$ if (i) it is divisible by $11$, (ii) it is divisible by $13$, (iii) it is divisible by $63$. (i) $23104*791=231 ...
0
votes
0answers
9 views

Where $a, b$ coprime, does $ax + b$ generate infinitely many 2-almost primes, infinitely many 3-almost primes, etc.?

I've seen various references to Dirichlet's theorem on arithmetic progressions claiming that where $a, b$ coprime, $ax + b$ not only generates infinitely many primes, but also infinitely many ...
0
votes
1answer
19 views

Show that $x_1x_2\cdots x_n (mod~ m)\equiv (x_1 (mod~m)\cdot x_2 (mod~m)\cdots x_n (mod~m))(mod~ m)$

Show that $x_1x_2\cdots x_n (mod~ m)\equiv (x_1 (mod~m)\cdot x_2 (mod~m)\cdots x_n (mod~m))(mod~ m)$ I know that $a\equiv b (mod ~ m)$, $c\equiv d (mod ~m)$ implies $ac\equiv bd (mod ~m)$ but how ...
1
vote
2answers
40 views

In how many ways I can write a number $n$ as sum of $4$ numbers?

The precise problem is in how many ways I can write a number $n$ as sum of $4$ numbers say $a,b,c,d$ where $a \leq b \leq c \leq d$. I know about Jacobi's $4$ square problem which is number of ways ...
4
votes
2answers
51 views

Use Wilson theorem to show that $63! + 1 \equiv 0 \mod ~ 71$

Use Wilson theorem to show that $63! + 1 \equiv 0 \mod ~ 71$. 71 is prime then Wilson theorem says that $(71-1)!+1=0 \mod ~ 71$ i.e $70!+1\equiv 0 \mod ~ 71$ then how to proceed further?
3
votes
7answers
78 views

What is the remainder when $6\times7^{32} + 7\times9^{45}$ is divided by $4$?

What is the remainder when $6\times7^{32} + 7\times9^{45}$ is divided by $4$ ? $7 \equiv 3 \pmod 4$ $7^2 \equiv 9 \pmod 4\equiv 1 \pmod 4$ $(7^2)^{16} \equiv 1^{16} \pmod 4$ i.e $7^{32} ...
1
vote
1answer
35 views

Find out a process to generate pairs of distinct positive integer $m$, $n$ with $\phi(m) = \phi(n)$.

Find out a process to generate pairs of distinct positive integer $m$, $n$ with $\phi(m) = \phi(n)$. Attempt: The pairs $m=1, ~ n=2$; $m=3, ~n=4$ satisfy the problem. But I need a ...
0
votes
0answers
33 views
+50

Understanding an example of a prime-like non-finite additive basis

In this answer on MO, the user Gene S. Kopp gives an example of a relatively "big" set $A\subset \mathbb{N}$ with relatively "small" gaps that fails to be an asymptotic finite basis. I'm having a ...
6
votes
1answer
35 views

counting function of system of equations and Circle method

I came up with the follwing question while looking on Davenport's book: Analytical Methods for Diophantine equations and Inequalities. When introducing the Circle method gives an example on how to ...
5
votes
6answers
405 views

Elementary number theory - prerequisites

Since summer comes with a lot of spare time, I've decided to select a mathematical subject I want to learn as much as possible about over the next three months. That being said, number theory really ...
1
vote
4answers
107 views

Given primitive solution to $\frac{1}{a}+\frac{1}{b}=\frac{1}{c}$, show $a+b$ is a perfect square [duplicate]

If $a,b,c$ are positive integers and $\gcd(a,b,c)$ is $1$. Given that $\frac{1}{a} + \frac{1}{b} = \frac{1}{c}$ then prove that $a+b$ is a perfect square. I was trying to get something useful from ...
2
votes
1answer
40 views

Why is $x^{100} = 1 \mod 1000$ if $x < 1000$ and $\gcd (x,1000) = 1$?

Let $U(1000) =$ the multiplicative group of all integers less than and relative prime to $1000$. "Show that for every $x \in U(1000)$ it is true that $x^{100} = 1 \mod 1000$." Been thinking ...
4
votes
0answers
25 views

Proving that there exist products of $a_k \equiv 1 \pmod {a_i}$ [closed]

Let $n>2$ be an integer. Prove that there exist numbers $a_1, a_2, \ldots ,a_n$ such that $$a_1a_2\cdots \widehat{a_i}\cdots a_n \equiv 1 \pmod{a_i}$$ for $i=1,2,3,\ldots,n$. Here ...
1
vote
3answers
58 views

Find solutions to $4x^2\equiv 1\pmod {29}$ [closed]

Find the solutions to the congruence: $$4x^2\equiv 1\pmod {29},\rm{ie},(2x)^2\equiv 1\pmod{29}$$
1
vote
1answer
39 views

Non real complex in metric completions of $\mathbb Q$

Process of completion of $\mathbb Q$ using the absolute value $|x|$ does not touch to the non-real complex numbers which are added to $\mathbb Q$ via extensions fields. However completion of $\mathbb ...
1
vote
1answer
31 views

Trouble with a proof: $(p^n - 1 , e)=1$ for $e\in \mathbb{N}$, p prime

I'm having trouble understanding a proof. The Lemma states: For every natural number $e$ there are infinitely many prime powers $q$ with $(q-1,e)=1$. The prove is as follows: Write $e=2^km$, m odd. ...
0
votes
0answers
48 views

Transitivity of discriminant for flat algebras

Let $A$ be an finite flat $R$-algebra and $A'$ be an finite flat $A$-algebra such that it is also finite flat as an $R$-algebra. Then we have a notion of discriminant ideals ...
4
votes
1answer
38 views

Furstenberg theorem and twin primes

The theorem of Furstenberg showing there exists infinitely many primes (and variants, including those stripping away the topological side of things) has been discussed several times on MSE, e.g. in ...
0
votes
0answers
21 views

square classes of quadratic extensions of 2-adic fialds

I have a question about square classes of quadratic extensions of 2-adic fields. I appreciate anybody help me to understand. Why all elements of $1+\mathfrak{p}^5$ are square in ...
2
votes
1answer
24 views

Bibilography: Riemann's hypothesis and positive semi-definite billinear forms

This is a bibliography request: I remember browsing through a book, some years ago, in a library, in which Riemann's hypothesis was proved over some type of fields (I cannot remember what type), the ...
-7
votes
2answers
66 views

Find the prime number [closed]

Find all prime numbers $a,b,c$ and positive integers $k$ satisfying the equation $a^2 + b^2 + 16c^2=9k^2+1$. I tried but I didn't came to any result.
2
votes
1answer
38 views

Does a bijection from the reals to the any binary form?

It is fairly simple to store all rational numbers in a binary format (not base 2) (a language composed of only 1s and 0s, no . marking) by simply storing one integer, a seperator, and another integer. ...
0
votes
0answers
23 views

How many tiles are Symmetrical? [closed]

We have a tape of type $1 * 2015$ had tile from tiles unit square in four different colors so as not exceed two tile of the same color (tile unit square, any tile from type $1*1$) How many tiles are ...
3
votes
2answers
76 views

$\log \log (\alpha)$ transcendental??

$\log \log (\alpha)$ transcendental?? ($\alpha$ algebraic $\neq 0$ and $1$) I supposed $\log \log (\alpha)=\beta$ , $\beta$ transcendental. Then $\log(\alpha)=e^{\beta}$ and it is know $e^{\beta}$ is ...
13
votes
1answer
156 views

Sum of Reciprocals of Primes in Imaginary Quadratic Field Diverges (2014 Miklós Schweitzer)

Problem 5 of the 2014 Miklós Schweitzer states: Let $\alpha$ be a non-real algebraic integer of degree two, and let $P$ be the set of irreducible elements of the ring $\mathbb{Z}[\alpha]$. Prove that ...
2
votes
0answers
33 views

Extend a map to a 1-cocycle

Let $\Gamma=PSL(2,\mathbb{Z})$ be the modular group with the usual presentation $\Gamma=\langle S,U,T|\ S^2=U^3=1, T=US\rangle$ where ...
0
votes
0answers
19 views

I attempt integrate another factor 2 in the definition of even perfect numbers

I use the method display by Florian in [1] (in true both statments of this problem are due to Florian at 99%) to compute from $\sigma(2n)-(\sigma(n)+\sigma(n))=2^p$ (where $\sigma$ is the sum of ...
0
votes
0answers
30 views

What is general Riemann's Hypothesis? [duplicate]

What makes it so important in analytic number theory?
1
vote
0answers
36 views

Why zero to the zero power is 1? [duplicate]

The google calculator say that $0^0=1$. I'm confused. It's well-known $0^0$ is undefined.
1
vote
1answer
35 views

$r! \equiv (−1)^k \pmod p$

Suppose that p ≡ 3 (mod 4) and $r = \frac {p-1}2$ Show that $r! \equiv (−1)^k \pmod p$ where k is the number of non-quadratic residues modulo p which are smaller than $\frac p2$ I know from ...
0
votes
1answer
49 views

Find all three numbers [closed]

Find all I three numbers, that can be divided to $11$, and the result equal to sum of square every digit from digits that number I tried but, I couldn't complete.
0
votes
2answers
40 views

Prove that $a+6b+4$ is full square. [closed]

Let $b=333...33, a=999...99$ where $a$ has $2n$ digit, $b$ has $n$ digit Prove that $a+6b+4$ is full square.
9
votes
1answer
134 views

Showing that the Prime Number Theorem is Plausible.

I have started to work through the course notes titled "Integers, Polynomials and Finite Fields" by Kenneth Davidson to keep me busy this summer, and there is a question in here This is an ...
5
votes
0answers
135 views

One-to-one correspondance between zeta zeros and the prime powers?

I have noticed an interesting property related to the Gibbs phenomenon for the Fourier transform of the zeta zeros in Riemann's explicit formula, namely that the rate at which $r\rightarrow 2 $ in the ...
0
votes
1answer
30 views

Show that $(r!)^2 ≡ (−1)^{r−1} \pmod p$ [duplicate]

I need to prove that if p is an odd prime and $r = (p-1)/2$ then $(r!)^2 ≡ (−1)^{r−1} \pmod p$ I think it has something to do with gauss's lemma ...
0
votes
0answers
36 views

How to write $n!=a^{\alpha_0}(a+1)^{\alpha_1}(a+2)^{\alpha_2}\cdots(a+r)^{\alpha_r}$?

We know that $n!=n(n-1)(n-2)\cdots3\cdot2\cdot1, n\in \mathbb N$. Now I am willing to write $n!$ as $a^{\alpha_0}(a+1)^{\alpha_1}(a+2)^{\alpha_2}\cdots(a+r)^{\alpha_r}$ where $a, r, \alpha\in \mathbb ...
1
vote
1answer
31 views

Prove/disprove the following asymptotic bound

Indicating with $p$ and $q$ prime numbers, is it true that for $x\rightarrow\infty$ $$ \sum_{\substack{p\leq x \\ p\equiv 1 ...
1
vote
1answer
43 views

show this equation has infinitely many postive rational numbers

show that : there exist infinitely many values of postive rational numbers $(a,b,c)$ such $$\begin{cases} a+b+c=3.78\\ abc=2 \end{cases}$$ ie: show this equation $$x^3-3.78x^2+kx-2=0$$ has ...
3
votes
3answers
49 views

Finding the last digit of $7^n$, $n\ge 1$.

I have noticed a cycle of 7,9,3,1. Meaning: $7^1\equiv 7\pmod {10}, 7^2\equiv 9\pmod {10}, 7^3\equiv 3\pmod {10},7^4\equiv 1\pmod {10}, 7^5\equiv 7\pmod {10}$ and so on. Therefore, if $n=4k+1$ the ...
4
votes
2answers
378 views

Is there any simple trick to solve the congruence $a^{24}\equiv6a+2\pmod{13}$?

Which of the following primes satisfy the congruence $$a^{24}\equiv6a+2\pmod{13}$$ 1) 41 2) 47 3) 67 4) 83 I am interested in Theorem statement, corollary, or Trick or Logic which solves this ...
1
vote
2answers
58 views

solve $x^2 \equiv 24 \pmod {60}$

I need to solve $x^2 \equiv 24 \pmod {60}$ My first question which confuses me a lot - isn't a (24 here) has to be coprime to n (60)??? most of the theorems requests that. what i tried - $ 60 ...
5
votes
0answers
35 views

Why are there palindromic subsequences at random among this sequence?

So I was thinking about the Goldbach conjecture and I rephrased it to myself as the following: Prove that every number lies halfway between two primes (or is itself prime.) Which is equivalent. ...
-4
votes
0answers
20 views

how do i find primitive roots of 13? [closed]

How many incongruent primitive roots does 13 have? Find a set of this many incongruent primitive roots mod 13.and what is the meaning of incongruent primitive roots?
3
votes
1answer
42 views

Does this product over the primes converge, and if so, to what?

I've been trying to play around with the product: $$\prod_{p \text{ prime}}\frac{1}{1-(-p)^{-1.5}}$$ Where the product runs over all the prime numbers. The product is similar in appearance to the ...
0
votes
2answers
37 views

Maximum number of digits in numbers between 0 to $n^2-1$ of base n

The number of digits in numbers between 0 and $n^2-1$ of base n is obtained by $\log_n(n^2) = 2\log_nn = 2$ But why log is ...
0
votes
1answer
20 views

Show how one can decrypt RSA message with e = 3 and $m<N^{1/3}$ without knowing the private key

Show how one can decrypt RSA message with e = 3 and $m<N^{1/3}$ without knowing the private key. I really don't know how to solve this one. we just learned about quadratic residues so i guess it ...
0
votes
2answers
68 views

Find $x+y+z$ from the equation

In the equation $$\dfrac{37}{13}=2+\dfrac{1}{x+\dfrac{1}{y+\dfrac 1z}}$$ find the value of $x+y+z$ How can I know the value of three variables while there is only one equation?
4
votes
3answers
55 views

Remainder when divided by 9

I'd like help with this question : What is the remainder when $$2^{2} + 22^{2} + 222^{2}+ \ldots + \underbrace{2222...22^{2}}_{49 \text{ times}} $$ is divided by $9$
0
votes
0answers
36 views

The amount of the third degree.

Often have to deal with such a cubic Diophantine equation. $$q(a^3+b^3)=t(x^3+y^3)$$ $q,t - $ are specified for the problem. Interesting - in all the values of the coefficient of solutions are ...