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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115 views

About a Sequence of Prime Numbers inspired by the Green Tao Theorem

I am learning math so this question may seem obvious. It is known from the Green Tao theorem that the sequence of prime numbers contains arbitrarily long arithmetic progressions. The Green Tao theorem ...
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2answers
42 views

question about number theory [duplicate]

If $p$ is a prime number and $p\mid a^p-b^p$. Then $p^2\mid a^p -b^p$.
1
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1answer
68 views

Are there prime gaps of every size?

Is it true that for every even natural number $k$ there exists some $n \in \mathbb{N}$ such that $g_n = p_{n+1} - p_n = k$? I don't know how to approach the problem at all, and in fact I don't even ...
1
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2answers
54 views

Solving $12x \equiv 20 \pmod{38}$

$12x \equiv 20 \pmod{38}$ $gcd(12,38)=2$ using Euclidean Algorithm. There is a solution since $2|20$. Use the Extended Euclidean Algorithm $2=12*-3 +1*38$ Then $20=2*10=12*-3*10+1*38*10$ so ...
1
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0answers
47 views

On the sum of the reciprocals of the zeros of $\zeta(s)$

It is well known that whenever $\rho$ is a nontrivial zero of the Riemann zeta function $\zeta(s)$, then $1-\rho$ is also a zero. But does the equality $\Re \sum_{\rho} \dfrac{1}{\rho} = \Re ...
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1answer
43 views

Modular Arithmetic - summing from 1 to a prime

Apologises for the vague title; I couldn't think of anything better to call it. I'm currently working on the following question: Consider the equation $\sum_{i=1}^{5} \frac{1}{i} = \frac{X}{5Y}$. ...
2
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1answer
51 views

Seeking more information regarding the “hybriation function.”

Definition 0. Given a pair of finite sets $Y$ and $X$, write $Y_X$ for the set of all collections $\mathcal{K}$ of functions $f : Y \leftarrow X$ that are closed under "hybridization", by which I ...
3
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2answers
58 views

Is the last digit of this number :$ {{4^4}^n}+1 $ always $7 $ for $n>1$ and could this be prime?

Some computations in wolfram alpha for $n=2,3,4,5 ,6$ showed that the last digit of this number $ {{4^4}^n}+1 $ for $n>1$ always $7$ . My question here :How do I know if it's last digit always ...
2
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1answer
39 views

What's so special about hyperbolic curves?

This is really a two-part question, but I would be happy to get an answer for either bit. By a hyperbolic curve as defined by e.g. Szamuely in Galois Groups and Fundamental Groups (p.137) I mean an ...
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2answers
28 views

Show that $c^2 \equiv -1 \pmod p$

Let $p \equiv 1 \pmod4$ be a prime. Write $p$ in the form $p=a^2+b^2$ where $a$ and $b$ are integers. Let $c \equiv ab^{-1} \pmod p$. Show that $c^2 \equiv -1 \pmod p$ $p = 4n+1$ where $n$ is an ...
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1answer
25 views

Dense on the unit circle

I am reading: "It is sufficient to show that the points $z_n = e^{2\pi in \xi}$ $\:\:n = (1, 2, 3...)$ are dense on the unit circle. ( $\xi$ is an irrational number)" How is this possible? Can ...
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1answer
30 views

$C(n)+P(n)+S(n)$ always composites?

Let $C(n)$ be the concatenation of first $n$ primes, let $P(n)$ be the product of first $n$ primes, and let $S(n)$ be the sum of the first $n$ primes. It is not surprising that $C(n) - P(n) - S(n)$ is ...
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0answers
51 views

No. of ways to Generate the String [duplicate]

I want to generate a binary string, such that number of occurrence of $00,01,10$ and $11$ are to be fixed. How can we find out the numbers of ways for given value. For example: number of occurrence ...
1
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0answers
83 views

Existence of a $G(x)$ that can generate all the even numbers?

Question This is a "spin-off" question of: Reformulation of Goldbach's Conjecture as optimization problem correct? I was wondering if a function existed such that: $$ G(x)^2 = ...
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0answers
15 views

Verification of Rogers-Ramanujan identities

In Hardy's book 'Ramanujan', section 6.8 on the Rogers-Ramanujan identities, it states: None of these proofs can be called both "simple" and "straightforward", since the simplest are essentially ...
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0answers
15 views

Instance of a Generalized Littlewood Conjecture

For real numbers $x$, let $\|x\|$ denote the distance from $x$ to the nearest integer. Define a function $\ell:\mathbb{R}^3\rightarrow\mathbb{R}$ by ...
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0answers
62 views

An analytic formula for the sum of the logs of primes.

I just read in Martin Klazar's Intoduction to Number Theory (page 53), that $\sum_{p\leq x} \log p - \log (p-1) = \log\log x + \gamma + O(1/\log x)$. Where $\gamma$ is the Euler-Mascheroni constant, ...
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1answer
32 views

On the sum of the logarithms of primes.

Let $p$ be a prime and $x$ be an integer. It is known that $\sum_{p\leq x} \log p = O(x)$, and i think this is equivalent to the Prime Number Theorem. ...
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5answers
68 views

How connect $x^2+xy+y^2$ to $j^3*4*n-27 = t^2$

$x^2 + xy + y^2 = (x^3 - y^3)/(x - y)$ Now let me show a subject not connected with above form (at least in some known way). By trying solve equation $1 \cdot 4 \cdot n - 27 = t^2$ ($n,t$ ...
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0answers
23 views

Solving the matrix in Quadratic sieve

I am trying to implement the quadratic sieve and I don't understand how to solve the matrix at the end. I will show you what I did and where I got stuck. So I am trying to factor $149 * 103 = 15347$ ...
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0answers
25 views

Are there clear, formal definitions for “terms” in subtraction operation?

I tutor children of all ages in Mathematics and I've noticed so many different words thrown around regarding binary operations, particularly with subtraction. For example, when working with a 2nd ...
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0answers
178 views

Reformulation of Goldbach's Conjecture as optimization problem correct?

Question I think I managed to reformulate a stronger version of Goldbach's conjecture as an optimization problem: $$ \frac{\partial F_n}{\partial a_n} = \frac{\partial F_n}{\partial \overline a_n} ...
2
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2answers
300 views

Is $(p-1)!+p$ a prime for every prime $p$?

Is $(p-1)!+p$ a prime for every prime $p$? It looks unlikely but I cannot get an example that will do. Also by trying to prove it by assuming a prime factor greater than $p$ and less than ...
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1answer
56 views

Find sum of two primes if their difference is equal to $3{ n }^{ 2 }-5n-1$

The difference between two prime numbers is equal to $3{ n }^{ 2 }-5n-1$. By using $n$, find the sum of them, where $n \in \mathbb{N}$. I didn't have any idea about how can I start to solve ...
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0answers
36 views

$p = a^2 + 2b^2$ Primes and congruences

Rational primes $p \geqslant 3$ of the form $p = a^2 + 2b^2$ factorize in $R=\mathbb{Z}[\sqrt{-2}]$ as a product of two irreducibles which are not associate. Such primes $p$ are $\equiv 1, 3 \pmod8$. ...
5
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0answers
58 views

Increasing function $f(x)$ such that $f(\gcd(x,y))=\gcd(f(x),f(y))$

This problem was largely inspired by this problem here. There were many counterexamples given to the problem, such as multiplicative function that maps primes to a permutation thereof. However, if ...
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1answer
35 views

Prove that there are as many quadratic residues mod p as there are quadratic non-residues mod p [duplicate]

I have found various proofs of this question usig primitive roots, but I want to prove it without using primitive roots! Here is my question again: Let $p$ be prime. Prove that there are the same ...
2
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1answer
26 views

On the divisors of $a^b + (\frac{a-b}2)^a + (\frac{b-a}2)^b+ b^a $

Let $a$ and $b$ be two odd natural numbers. Show that $\frac{a+b}{2}$ divides $$a^b + \left(\frac{a-b}2\right)^a + \left(\frac{b-a}2\right)^b+ b^a$$
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20 views

Generalizing Landau-Ramanujan Theorem for sum of three squares

Let $S_{3}(x)$ denote the number of positive integers not exceeding x which can be expressed as a sum of three squares. Can we find an asymptotic formula for $S_{3}(x)$, maybe using Landau-Ramanujan ...
5
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4answers
226 views

Which is bigger: $(n!)^{n!}$ or $(n^{n})!$? [closed]

To be honest I haven't spent a whole lot of time thinking about this other than the drive back home, and I won't have much time to think about it for a while due to shit-happening. So i thought I'd ...
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0answers
68 views

Simple logical proof of Fermat's Last Theorem [closed]

My interest in the Fermat Conjecture (FC,) began as an interest in the Pythagorean theorem. I wasn't looking for integer solutions of n>2. I was more interested in the fact that odd integer values of ...
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2answers
34 views

Prove the if $a\nmid x$ and $a\nmid y$ then $a\nmid xy$

I need help proving that if $a\nmid x$ and $a\nmid y$ then $a\nmid xy$. I want to do this preferably without a counterexample. I already know that this is False, but I want to know how.
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1answer
38 views

finding generators for spaces of half-integral weight modulars of level 8

I'm trying to realize the spaces of half-integral weight modular forms for $\Gamma_{0}(8)$ as the spaces of polynomials in some modular forms of level 8. For every integer $k$, it is known that every ...
2
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0answers
38 views

Find all integer sets $a,b$ where $\gcd(a^2+a+1,b^2+b+1)=3$

How does one find all integer sets $a,b$ where $gcd(a^2+a+1,b^2+b+1)=3$? It appears that for $gcd(a^2+a+1,b^2+b+1)=3$ to be true, then $a\equiv b \equiv 1 \pmod 3$. Also, if $(a^2-a+1,b^2+b+1)=1$, ...
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2answers
82 views

Find all $f(x)$ such that $f(gcd(x,y))=gcd(f(x),f(y))$

How does one find all $f:\mathbb {Z} \rightarrow \mathbb {Z}$ that satisfies the following: $$f(gcd(x,y))=gcd(f(x),f(y))$$ I had suspected that there would be some results concerning this functional ...
2
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2answers
24 views

How can I use primitive roots to prove that there are the same number of quadratic residues as non-residues?

I am given the fact that the Legendre symbol, $\left(\frac{\omega}{p}\right) = -1$. How can I use this to prove that there are as many quadratic residues as quadratic non residues modulo p? Here, $p$ ...
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1answer
39 views

Automorphic numbers exercise

The number 9376 has the peculiar self-reproducing property that $$9376^{2}=87909376$$ How many 4-digit numbers x satisfy the equation $$x^{2}\equiv x\bmod10000?$$ How many n-digit numbers x satisfy ...
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1answer
54 views

The ring $R_{p}$

Let p be a prime number in $\mathbb{Z}$. Let $R_{p}$ be the ring $R_{p} : = \{x \in \mathbb{Q} : ord_{p}(x) \geq 0\}$. Show that x is a unit of R if and only if $ord_{p}(x)= 0$. I'm not sure how to ...
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1answer
28 views

examples of unramified extensions of $\mathbb{Q}_p$

For every local field $K$ and natural number $n$ coprime to $K$'s residue characteristic, there is a unique unramified extension $L/K$ of degree $n$. Let's take $K=\mathbb{Q}_p$. What are some ...
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3answers
132 views

Show that $504 \mid (n^9 − n^3 )$ for any integer $n$ [closed]

Not sure what to do / how to start this... I have equcation of 504 is: $2 \cdot2 \cdot 2 \cdot 3 \cdot 3 \cdot 7$
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0answers
29 views

On primes between consecutive $n-th$ powers. [duplicate]

The Opperman conjecture, is the statement that for every integer $x\geq 2$, there always exists a prime btween $x^2$ and $(x+1)^2$. How about for every integer $n\geq 3$, is there always a prime ...
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1answer
26 views

Logic behind problem related to LCM.

This might be a stupid question, but I am not able to find any good explanation of this on the internet. Suppose there are two numbers $28$ and $32$. Now, we need to find the smallest number that ...
4
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1answer
59 views

Is it possible to compute the $\limsup$ of $x_n$ where $x_n$ is the $n^{th}$ digit of $\pi$?

I'm curious about something. Let $x_n$ represent the $n-$th digit of $\pi$ i.e. $\pi=3.1415$ and thus $$x_0=3,\quad x_1=1,\quad x_2=4,\quad x_3=1,...$$ Is it possible to determine ...
4
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0answers
62 views

What is the smallest prime factor of the number $14^{14^{14}}+13\ $?

What is the smallest prime factor of the number $$N\ :=\ 14^{14^{14}}+13\ ?$$ The number of digits of $N$ is $12,735,782,555,419,983$ (The number of digits of $N$ has itself $17$ digits). The first ...
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1answer
41 views

Which two integers produce random infinite sequence when the largest divided by the smallest?

Which two integers produce random infinite sequence when the larger one is divided by the smaller one? For instance, $\frac{920}{33}= 27.8787878787...$, is not a random sequence.
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2answers
32 views

Find all $n$ such that $m = an$ or $m =\dfrac{n}{a}$

$a$ is the 1st digit (from the left) of a $3$-digit number $n$. We get the number $m$ by removing a from $n$ and putting it on the right of the unit-digit. For example, the number $123$ becomes $231$. ...
0
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1answer
26 views

What is an upper bound for number of prime powers and semi primes in the interval $[n^2+1,n^2+n]?$

What is an upper bound for number of prime powers in the interval $[n^2+1,n^2+n]?$ What is an upper bound for number of square free semi primes in this interval$?$
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1answer
79 views

The first $4$ primes $p$ for which $15347$ has a square root mod $p$ are $2, 17, 23,$ and $29$

I am reading about Quadratic Sieve article in wiki and I don't understand the sieve part. The article says: The first $4$ primes $p$ for which $15347$ has a square root mod $p$ are $2, 17, 23,$ ...
1
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1answer
13 views

When does:$(p^y+1 )\bmod (p^x+1)=0 $ if $(y,x)=1$ and $p $ is a prime number?

I'm interesting to look the solution of this equation :$$(p^y+1 )\bmod (p^x+1)=0 $$ at a least to see an example of the two coprime $y, x$ for which $(p^y+1 )\bmod (p^x+1)=0 $ but i don't succed , ...
3
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0answers
18 views

Twisted Kloosterman Sums

A twisted Kloosterman sum is a character sum of the form $$S(\chi, \psi, \eta)=\sum_{t\in (\mathbb{F}_q)^{\times}} \chi(t) \psi(t) \eta(t^{-1}).$$ Here, $\chi$ is a multiplitive character of ...