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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2
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0answers
18 views

Can the determinant of an integer matrix with $k$ given rows be the gcd of the determinants of the $k\times k$ minors of those rows?

I'm interested if the following is true: Let $n\geq k\geq1$ be integers and let $$A=\begin{pmatrix}a_{11}&\cdots&a_{1n}\\ \vdots&\ddots&\vdots\\ ...
0
votes
1answer
39 views

how to solve 192-2a^2-a=m(6a+1)?

how to solve $192-2a^2-a=m(6a+1)$ ? or written as $(192-2a^2-a) \equiv 0$ (mod $6a+1$) how to calculate the integer values of $a < 41$ ? thanks to understand that serving: ...
2
votes
2answers
42 views

Is the sequence $\{0,2,6,12,20,30,…,n(n+1)\}$ admissible for every natural $n$?

Look here : https://en.wikipedia.org/wiki/Prime_k-tuple for the definition of an admissible sequence. I wonder if the sequence of differences of primes can be $\{0,2,4,6,8,...,2n\}$ for every ...
1
vote
1answer
18 views

raising elements of profinite groups to $p$-adic powers

Let $\widehat{F_2}$ be the profinite free group of rank 2, and let $\widehat{\mathbb{Z}}$ be the profinite completion of $\mathbb{Z}$, and $\widehat{\mathbb{Z}}^\times$ its group of units. For ...
3
votes
0answers
58 views

Thue equation $ x^4 - 6 x^3 y - x^2 y^2 + 6 x y^3 - y^4=-1 $

I need help solving the Thue equation $ x^4 - 6 x^3 y - x^2 y^2 + 6 x y^3 - y^4=-1 $. It can be written as $ x(x-y)(x+y)(x-6y) = (y-1)(y+1)( y^2 +1) $. From this I found 8 solutions ...
2
votes
1answer
25 views

Algebraic integers of $\mathbb{Q}(\sqrt{m})$ for $m$ a squarefree integer

I'm currently reading Marcus' "Number Fields," and I'm having difficulty proving the following result: Corollary 2.2: Let $m$ be a squarefree integer. The set of algebraic integers in the quadratic ...
3
votes
2answers
56 views

Prove that for every natural number $n > 2$ there is a prime number between $n$ and $n!$

So I have already read this page with the solution: For all $n>2$ there exists a prime number between $n$ and $ n!$ Now I was able to reason that $p < n!$ Because I was given the hint that ...
2
votes
1answer
38 views

Numbers relative to their sum of Divisors

Define the D-Ratio as the ratio of a natural number $n$ as: the sum of $n$'s Divisors, excluding 1 and $n$ divided by $n$ itself. [Thus the D-Ratio of $24$ is $$\frac{2 + 3 + 4 + 6 + 8 + 12}{24} = ...
-2
votes
1answer
29 views

Two Vertical Lines

What does two single vertical lines mean in math. I am thoroughly confused by this question: What describes |3/1|? Use all that apply.
1
vote
1answer
34 views

Integral of polynomial related to prime divisors

Given the following integral $I_{m,n}=\int_{0}^{1}(1-x^n)^m \mathrm{d}x$. Prove that for any fixed $n$ and for any $m$ $I_{m,n}$ is a rational number and when written in the form $\frac{p}{q}$ with ...
-2
votes
3answers
47 views

Poof that: 1. $gcd(a,b)=lcm(a,b)$ If and only if $a=±b$ 2. if $k>0$, then $lcm(ka,kb)=klcm(a,bk)$ 3. if $m$ is multiple of $a,b$, then $lcm(a,b)$

Let a,b any integers, with $a,b≠0$ Poof that: $gcd(a,b)=lcm(a,b)$ If and only if $a=±b$ if $k>0$, then $lcm(ka,kb)=klcm(a,bk)$ if $m$ is multiple of $a,b$, then $lcm(a,b) | m$ i know the ...
0
votes
2answers
51 views

Prove that $\mathbb{Z}[i]$ consists precisely of the elements of $\mathbb{Q}(i)$ which satisfy $x^2 + ax + b=0$, $a,b \in \mathbb{Z}$

I was reading Neurkich's "Algebraic Number Theory" and there was a proof in it that makes no sense. Proposition 1.5: $\mathbb{Z}[i]$ consists precisely of the elements of the extension field ...
5
votes
1answer
51 views

Missing values of the ratio $\frac{(x+y+z)^2}{x^2+y^2+z^2}$

Let $x,y,z$ be some positive integers. Is it true that we cannot find any positive integer $n$ for which $$ \frac{(x+y+z)^2}{x^2+y^2+z^2}=1+\frac{2}{3n}\,\,? $$
2
votes
2answers
72 views

If $x^{100}$ is 31 digit number Then $x^{1000}$ contains how many digits.

If $x^{100}$ is 31 digit number Then $x^{1000}$ contains how many digits. Our Approach: $10^1$ has $2$ digits = $10$ $10^2$ has $3$ digits = $100$ $10^3$ has $4$ digits = $1000$ $10^{30}$ has ...
0
votes
0answers
31 views

Find the value of $b(100)-b(97)-b(96)$ if $b(0)=1$ with given conditions

If for any series $b(n)=2b(n-1)$ when $b(n)$ is odd number and $b(n)=b(n-1)$ if $b(n)$ is even number. then Find the value of $b(100)-b(97)-b(96)$ if $b(0)=1$ Our Approach: I could not ...
2
votes
0answers
23 views

What will be the last number of the set B in which a set B={$2$,$3$,$5$,$6$,$7$,$10$,_ ,_ ,_______} contain $300$ nos.

A set B={$2$,$3$,$5$,$6$,$7$,$10$,_ ,_ ,_______} contain $300$ nos. in which squares and cube of the no. are eliminated. then what will be the last number of the set B? Our Approach: As we have ...
1
vote
1answer
52 views

Smallest twin-prime-pair exceeding $10^{1000}$

I found the twin-prime-pair $$\large 10^{1000}+9705092\pm 1$$ with PARI/GP. Is this the smallest twin-prime above $10^{1000}$ ? A general question to the search of twin primes : The prime number ...
0
votes
0answers
21 views

Issue with modular arithmetic problem

So I have a problem with this question I was doing. I found that $94^6+32\cdot28^6$ is divisible by 2013, using a calculator. Since 61 divides 2013, 61 also divides $94^6+32\cdot28^6$. However, i ...
2
votes
2answers
62 views

How do I find(isolate) the n-th prime number?

So I wanted to solve this SPOJ problem and I did some research about finding the n-th prime number. This formula came across and it stated that the n-th prime must be in this range: $n \ln n + ...
6
votes
1answer
57 views

Can the determinant of an integer matrix with a given row be any multiple of the gcd of that row?

Let $n\geq2$ be an integer and let $a_1,\ldots,a_n\in\mathbb Z$ with $\gcd(a_1,\ldots,a_n)=1$. Does the equation ...
0
votes
0answers
54 views

why $\frac{a}{b}\pmod p=\frac{a\pmod p}{b\pmod p}$

It is said this following is theorem? what's this name? and How to prove it? Thanks show that $$\dfrac{a}{b}\pmod p=\dfrac{a\pmod p}{b\pmod p},a,b\in N^{+},(a,p)=1,(b,p)=1$$
3
votes
3answers
193 views

What is the Max value of n when 185! is divided by (189^n) will give an Integer Value?

What is the Max value of n when $185!$ is divided by $(189^n)$ will give an Integer Value? Options are a) $91$ b) $30$ c) $36$ d) $24$ MyApproach: $189$=$3^3$ . $7$ When $185$/$3$=$61$ ...
1
vote
0answers
26 views

Is there a tighter approximation for the least prime gap of a given length?

This link https://primes.utm.edu/notes/gaps.html gives a definition of the maximal gaps. For a number $g$ , $p(g)$ is the smallest prime $p$ followed by at least $g$ composites. The estimate is ...
0
votes
2answers
33 views

$z=100^2-x^2$. Then, how many values of $x,z$ are divisible by $6$?

$z=100^2-x^2$. Then, how many values of $x,z$ are divisible by $6$? My approach: For $x=1$, $z$ is not divisible by $6$. For $x=2$, $z$ is divisible by $6$. For $x=3$, $z$ is not divisible by ...
0
votes
0answers
56 views

If a, b are positive integers and $(ab - 1) \mid (a² + b²)$ then prove that $q = \frac{a² + b²}{ ab - 1} = 5$. [on hold]

If a, b are positive integers and $(ab - 1) \mid (a² + b²)$ then prove that $q = \frac{a² + b²}{ ab - 1} = 5$. I know it has something to do with number theory
2
votes
4answers
41 views

Intuitive explanation for p ∨ q → r ≡ ( p → r) ∧ (q → r)

Although, it is possible to prove the above equivalence using truth tables, I don't know how to prove it without using truth tables.Can someone explain it in plain english?
2
votes
3answers
41 views

Fastest way to perform this multiplication expansion?

Consider a product chain: $$(a_1 + x)(a_2 + x)(a_3 + x)\cdots(a_n + x)$$ Where $x$ is an unknown variable and all $a_i$ terms are known positive integers. Is there an efficient way to expand this?
4
votes
4answers
70 views

Books on Prime numbers

I am a graduate student and have just finished Burton's book on number theory. Now I want to read further on prime numbers. Does anyone have any suggestion?
-1
votes
1answer
38 views

LCM of randomly selected integers

What is the expected LCM of 21 randomly selected positive integers under 10000000? How would someone even approach this problem? EDIT: The positive integers are chosen with replacement.
2
votes
2answers
35 views

Can this congruence be simplified?

$$p(p+1) \equiv -q(q+1) \bmod pq$$ Can this be reduced to an easier format?
3
votes
1answer
36 views

Large regions of the plane $(x,y) \in \mathbb{Z}^2$ with no relatively prime points: $\mathrm{gcd}(x,y) > 1$

For any $\ell > 0$ can you find $M, N$ such that $ \boxed{\mathrm{gcd}(x,y) > 1}$ for all $x \in [M, M+\ell]$ and all $y \in [N, N+\ell]$ ? This is related to the statement that the set of ...
4
votes
1answer
38 views

Pentagonal Numbers

I recently was passing some time on Project Euler, when I came across this question. It deals with finding Pentagonal Numbers $P_j$ and $P_k$ such that $P_j+P_k$ and $P_j-P_k$ are also pentagonal ...
-1
votes
1answer
25 views

Is there a number which describes the approaching ratio of twin primes to other primes? Or a formula for the change in density of twin primes?

Could someone shed some light on what we know about the density of twin primes? I find that it seems to be empirically true that the density of prime gaps increases as $\log(x)$ does for any gap. ...
1
vote
0answers
42 views

Two irrational numbers are congruent iff the tails of their infinite continued fractions eventually coincide

We say that a real number $\alpha$ is $congruent$ to real number $\beta$ if there exist integers a, b, c and d with ad-bc=+1 or -1 and such that $$\alpha=\frac{a\beta +b}{c\beta+d}$$ I need to prove ...
5
votes
0answers
45 views

Integers which are the sum of non-zero squares

Lagrange's four-square theorem states that every natural number can be written as the sum of four squares, allowing for zeros in the sum (e.g. $6=2^2+1^2+1^2+0^2$). Is there a similar result in which ...
5
votes
0answers
80 views

Diophantine equation: $13^x+3=y^2$

$$13^x+3=y^2\iff \left(4+\sqrt{3}\right)^x\left(4-\sqrt{3}\right)^x=\left(y+\sqrt3\right)\left(y-\sqrt3\right)$$ $\gcd\left(y+\sqrt3, y-\sqrt3\right)=1$, therefore ...
0
votes
1answer
56 views

How to compute an integral?

I am reading the lecture notes. I am trying to understand the prove of Lemma 0.0.1.1 on page 4. From line 3 to line 4 in the proof of Lemma 0.0.1.1., how to prove that $$ \int_{F^{n-1}} ...
4
votes
1answer
59 views

Each number in $S\subseteq \{1,\ldots,2n\}$ does not divide another one, with $|S|= n$. In how many ways?

Let $f(n)$ be the number of subsets $S\subseteq \{1,2,\ldots,2n\}$ such that $|S|=n$ and $a$ does not divide $b$ whenever $a,b \in S$ are distinct. Can we evaluate $f(n)$, at least asimptotically? ...
0
votes
1answer
50 views

Roots of $\Phi_{31}(x)$ as roots of unity.

Let $\Phi_{31}(x)$ be the $31$-cyclotomic polynomial. I want to show that $\Phi_{31}(x)$ is the product of six irreducible quintic factors in $\mathbb{F}_2$. I am running into difficulties ...
0
votes
0answers
17 views

Composition of polynomial and multiplicative is multiplicative .

I made the following problem a while ago but I can't solve it (also I don't think it's extremely hard ) : Let $f$ be a non-constant completely multiplicative function over $\mathbb{Z}$ . Assume ...
12
votes
1answer
78 views

parallel resistors

Consider the set $E_b = \left\{1, 1.2, 1.5, 1.8, 2.2, 2.7, 3.3, 3.9, 4.7, 5.6, 6.8, 8.2\right\}$. This is our base set. Let's define the set $E$ as follows: $$ E = \left\{ 10^k e \mid k=0,1,2,\ldots, ...
1
vote
0answers
14 views

Symmetry of Hecke L-function zeroes

For the Riemann zeta function, it is known by the functional equation and $\zeta(s)=\overline{\zeta(\bar s)}$ that the zeroes of $\zeta(s)$ are symmetric about the critical line $1/2$ and the real ...
2
votes
1answer
20 views

Is $a+r \cdot b$ an uniformly random value when $a,b$ are fixed and $r$ is random value?

Imagine we have two fixed values $a,b \in \mathbb{Z}_p$ and a uniformly random value $r\leftarrow \mathbb{Z}_p$, for large prime number $p$. Question: Is $v=a+b\cdot r$ an uniformly random value in ...
3
votes
0answers
31 views

Unit group of an imaginary quadratic ring

Let $R$ be an imaginary quadratic ring. Then, the unit group $R^{\times}$ is finite. To prove this, I worked with normal forms, algebraic integers and the fact that $R \not \subset \mathbb{R}$. But I ...
1
vote
4answers
67 views

Solving $a!+b!+c!=3^d$

The question is to find all tuples $(a,b,c,d)$ of natural numbers $c\geq b$ and $ b \geq a $ and $a!+b!+c!=3^d$. I am finding difficulty in establishing relation between $a$, $b$, $c$, and $d$. I see ...
2
votes
2answers
40 views

How many pairs of natural numbers $(x,y)$, satisfy the equation $\space xy=x+y+\gcd(x,y)$.

How many pairs of natural numbers $(x,y)$, satisfy the equation $\space xy=x+y+\gcd(x,y)$. You may assume that $x≤y$.
4
votes
1answer
39 views

Example of an nonidentity element in the kernel of the map.

This question is related to my previous question here. Let $n, m > 1$. The map $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/m\mathbb{Z}$, of reduction mod $m$, induces a group homomorphism $F: ...
3
votes
0answers
24 views

Kronecker symbol vs. Koblitz symbol

In Koblitz, Introduction to Elliptic Curves and Modular Forms on page 188 it is defined $$\left( \frac{-1}{j}\right)=0$$ in case $j$ is even. Apart from that definition $\left( \frac{c}{d}\right)$ is ...
1
vote
0answers
42 views

Does anyone recognize this graph?

It's a plot of the following: Let $$f_{(n)} = \frac{np_n}{(p_1 + \ldots + p_n)}$$ so that $$g_{(n)} = \left|\space f_{(n)} - f_{(n-k)}\right| $$ where $n > k$ and $k = 5$ in this example. For ...
0
votes
1answer
45 views

Show that there exists $s, t \in S$ such that $\gcd(s, t)$ is a prime

Let $S$ be a set containing finitely many positive integers greater than 1 with property: for all $n \in \mathbb{Z_+}$, there exist $s \in S$ such that $\gcd(s, n) = 1$ or $\gcd(s,n) = s$. Show that ...