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.

learn more… | top users | synonyms (1)

-1
votes
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
vote
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 = ...
0
votes
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 ...
0
votes
0answers
16 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 ...
0
votes
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, ...
0
votes
1answer
35 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. ...
1
vote
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 $1 \cdot 4 \cdot n - 27 = t^2$ ($n,t$ ...
0
votes
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 $149 * 103 = 15347$ ...
0
votes
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 ...
0
votes
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
votes
2answers
301 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 ...
0
votes
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 ...
0
votes
0answers
49 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
votes
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 ...
0
votes
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
votes
1answer
27 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 $$a^b + \left(\frac{a-b}2\right)^a + \left(\frac{b-a}2\right)^b+ b^a$$
1
vote
0answers
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
votes
4answers
230 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 ...
-5
votes
0answers
70 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 ...
0
votes
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.
0
votes
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
votes
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 $(a^2-a+1,b^2+b+1)=1$, ...
7
votes
2answers
84 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
votes
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, $p$ ...
0
votes
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 ...
1
vote
1answer
63 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 ...
1
vote
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 ...
-1
votes
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: $2 \cdot2 \cdot 2 \cdot 3 \cdot 3 \cdot 7$
0
votes
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 ...
1
vote
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
votes
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
votes
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 ...
0
votes
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.
1
vote
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
votes
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$?$
4
votes
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
vote
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
votes
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 ...
0
votes
1answer
27 views

Find all pairs of $decianimales$

An ordered pair $(a, b)$ of positive integers is called $decianimal$ when $\dfrac{1}{a}+\dfrac{1}{b} $ is equal to a decimal fraction $\dfrac{m}{10}$ with $gcd(m,10)=1$. Find all pairs ...
2
votes
0answers
38 views

How to generate primitive solutions to the equation $a^3 + b^3 = c^2$

The solution for this is that we are supposed to pick numbers x and y, then we can substitute them in the equation and obtain some z, which we then multiply the left side of the equation with to ...
5
votes
3answers
97 views

Find the third rational point on the curve: $y^2 = x^3 + 8$

I am trying to find a third rational point on the curve $y^2 = x^3 + 8$ According to the my professor's solution, the idea is to find two rational points then solve for the third point. These are ...
1
vote
2answers
44 views

Geometric Interpretation of Complex Algebraic Proof of Sum of Squares Statement

When I see answers regarding proofs such as the one mentioned here, it seems that there is a considerable diversity of ways to attempt to look at this proof. Similarly, although this sum of squares ...
4
votes
2answers
54 views

What integers are coprime to the first $x$ prime numbers?

I have noticed that there is a very specific pattern to numbers that are coprime to $2$, it is simply all of the odd numbers. More specifically, it is in the following pattern, where $n$ is an ...
3
votes
2answers
40 views

Primitive Root Mod P

I have been able to answer all of the parts to the question apart from part (v). Any tips on how? I assume I have to show that $ m=2^n $, but I am unsure as to how. I can't imagine it is very ...
0
votes
1answer
26 views

Can you square (or exponentiate to any power) both sides of a modular equation?

If $ a \equiv b \;\;(mod\;p) $ then by definition this means $p | (a - b)$. Now for $n = 2$ we would have $ a^2 \equiv b^2 \;\;(mod\;p) $ or $p | a^2 - b^2$ and $p | (a - b)(a + b) $. Now this will ...
0
votes
1answer
30 views

multiples of subset of $\mathbb{N}$

Suppose that $ A_1\cup \dotsm A_n=\mathbb{N}$ is a partition of $\mathbb{N}$ into disjoint subsets. Is it true that there is an integer $1 \leq k \leq n$ such that the set $A_k\cap 2A_k$ is infinite?
0
votes
2answers
31 views

RSA Encryption Original Primes $p$ and $q$

I am well aware of the math behind the RSA encryption system, and why it works. The bank, for example, publishes a pair of numbers $(e,n)$ which are used for encryption by the customers. The bank then ...
3
votes
1answer
47 views

2 is a square modulo $p$ if and only if $p \equiv \pm 1 \pmod 8$

2 is a square modulo $p$ if and only if $p \equiv \pm 1 \pmod 8$ The indication for this exercise was to consider $\alpha$ such that $\alpha^8 = 1$ ($\alpha$ is eigth root of the unity) in a ...
4
votes
3answers
44 views

What is the extraneous solution of $\sqrt a=a-6$?

What is the extraneous solution of $$\sqrt a=a-6$$ The roots are $9$ and $4$. So I'm assuming that $4$ is the extraneous solution because when you plug it in to the equation you wind up with $2=-2$. ...
2
votes
1answer
42 views

Density of numbers with exactly $n$ distinct prime factors in $\mathbb{N}$

It is quite well known that the density of the primes in $\mathbb{N}$ is $0$, that is, $$\lim_{n\to\infty}\frac{|\{p\mid p\leq n, p \text{ prime}\}|}{|\mathbb{N}_{\leq n}|}=0$$ It is less well-known, ...