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
61 views
+50

Geometric average of prime gaps?

The aritmetic average of prime gaps around $x$ is $\ln(x)$. What is the geometric average of prime gaps around $x$ ? Does that strongly depend on the conjectures about the smallest and largest gap ...
5
votes
2answers
68 views

How to decide which moduli to check when solving a “polynomial” congruence?

Consider the following problem: Find all integer solutions to $y^2 = x^5 - 4$. The solution goes something like – check modulo 11, where $x^5 \equiv 0, \pm 1$, and then check cases to arrive at ...
1
vote
2answers
20 views

Number of solutions to congruences

Is there any general form to determine the number of non-congruent solutions to equations of the form $f(x) \equiv b \pmod m$? I solved a few linear congruence equations ($ax \equiv b \pmod m$) and I ...
1
vote
1answer
14 views

Show the number of quadratic residues $a$ modulo $p$ with $1\leq a\leq p-1$ is $(p-1)/2$

Question: When $p$ is an odd prime, show that the number of quadratic residues $a$ modulo $p$ with $1\leq a\leq p-1$ is $(p-1)/2$ Answer: From Euler's criterion $\left(\frac{a}{p}\right)\equiv ...
0
votes
0answers
13 views

Show $\text{Gal}(K_\infty/\mathbb Q)\cong \mathbb Z_p^{\times}$

Let $\zeta_{p^n}$ be the primitive $p^n$-th root of unity where $p$ is a prime and $K_n=\mathbb Q(\zeta_{p^n})$ the $p^n$-th cyclotomic field. Let $K_\infty=\bigcup K_n$. Could someone give a proof ...
0
votes
1answer
13 views

Solving recurrence with moebius inversion

what's up folks? I'm solving the red book of math problems, problem 16 which is to solve the following recurrence relation: $\sum_{k=1}^n {n \choose k} a(k) = \frac{n}{n+1}$ PS: ${n \choose k} = ...
1
vote
2answers
48 views

Generalized Pell equation $x^2-2y^2=49$

How can I find the integer solutions of the generalized Pell equation $x^2-2y^2=49$ I have an algorithm, but this only works if $D\ge r^2+\frac{\left(|r|+1\right)}2$, (where $x^2-Dy^2=r^2$). ...
0
votes
1answer
32 views

Show that $f(a)$ converges after some point

There is a row of 1000 integers. There is a second row below, which is constructed as follows. Under each number $a$ of the first row, there is a positive integer $f(a)$ such that $f (a)$ equals ...
2
votes
2answers
24 views

if $P$ is a prime ideal of $O_K$, then $O_K/P$ is finite

let $P$ be a non-zero prime ideal of $O_K$, where $K$ is a number field(i.e. the degree $[K:\mathbb{Q}]$ is finite) then $O_K/P$ is finite. I'm working through a proof for this claim, however there is ...
15
votes
2answers
908 views

So can anybody indicate whether it is worthwhile trying to understand what Mochizuki did?

So I am looking at some math stuff and I start looking at the abc-conjecture. Naturally I run into the name Mochizuki and so start trying to see what he did. Well, he is starting look like another ...
0
votes
0answers
19 views

Finding the m-th coefficient of an equation

Does anyone know how to find the m-th coefficient of $-t=(3+2\sum_{k=1}^{\infty} \frac{t^{2k}}{(2k)!})(\sum_{n=0}^{\infty} D_n \frac{t^n}{n!})$? The answer is: $0=3 \frac{D_m}{m!}+2\sum_{k=1}^{m/2} ...
0
votes
1answer
1k views

finding the greatest perfect square dividing an integer

how can we find the greatest integer which is a perfect square and which divides an integer? I believe factorisation can be used here but am not sure how to get the result out of it for all prime, ...
4
votes
4answers
130 views

How to solve $x^3\equiv 10 \pmod{990}$? [closed]

How to solve $x^3\equiv 10\pmod{990}$? It has 3 solutions: 10, 340, 670 (WolframAlpha).
0
votes
1answer
28 views

Is $\pi(y)\pi(x+k)\ge\pi(x)\pi(y+k)$?

Is it true that for some fixed $k\ge2$ and for all sufficiently large $x$ and $y$ with $y\ge x$ we have, $$\pi(y)\pi(x+k)\ge\pi(x)\pi(y+k)$$ where $\pi(x)$ is the prime counting function. I am ...
5
votes
0answers
24 views

$\sum_{n=1}^{\infty}\frac{a_n}{10^n}$ where $\{a_n\}_{n=1}^{\infty}$ is a sequence in the ten digits {0, 1, 2 , 3 , 4 , 5 , 6 , 7 , 8 ,9}

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence in the ten digits {0, 1, 2 , 3 , 4 , 5 , 6 , 7 , 8 ,9} And consider the sum $\sum_{n=1}^{\infty}\frac{a_n}{10^n}$ $\in$ $[0,1]$ What characteristics of ...
0
votes
2answers
25 views

If $ p $ is an odd prime and $ D $ an integer not divisible by $ p $, show that $ x^2 - y^2 \equiv D ~ (\text{mod} ~ p) $ has $ (p - 1) $ solutions.

I am supposed to have proved the following congruence identity: $$ 1^{n} + 2^{n} + \cdots + (p - 1)^{n} \equiv 0 ~ (\text{mod} ~ p). $$ This is apparently meant to help me solve the problem stated in ...
0
votes
0answers
16 views

Algebra inequality problem

Suppose: $ x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 1$ , and $x_1x_3x_5 + x_2x_4x_6 \ge \dfrac {1}{540} $ and $\dfrac{p}{q}$ is the maximum possible value of $x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_6 ...
0
votes
1answer
16 views

Does $cN \leq \left|\sum_{n= 1}^{N}f(n)\right|$ imply $|f(n)| \geq c/2$ for many $n$?

Let $f: \mathbb{N}\rightarrow \mathbb{R}$ be a function potentially taking negative values. Suppose I knew that $$cN \leq \left|\sum_{n= 1}^{N}f(n)\right|$$ for some absolute constant $c$. Then by the ...
3
votes
1answer
107 views
+500

Mordell-Weil rank in elliptic surfaces

Suppose that an elliptic smooth K3 surface $X$ defined over a number field $k$ has arithmetic Picard rank $r$ and assume that it is equipped with a $k$ fibration over $\mathbb{P}^1$ that has a section ...
0
votes
0answers
21 views

Prove that if $x^3 + y^3 = z^3$, and $x$, $y$, $z$ are quadratic integers in $\mathbb Q[\sqrt{−3}]$, then $λ$ must divide one of $x$, $y$, or $z$.

Let $λ = (3 + \sqrt{−3})/2 ∈ Q[\sqrt{−3}]$. Prove that if $x^3 + y^3 = z^3$, and $x$, $y$, $z$ are quadratic integers in $\mathbb Q[\sqrt{−3}]$, then $λ$ must divide one of $x$, $y$, or $z$.
1
vote
1answer
24 views

Prove by counterexample that $\gamma$ and $\delta$ are not necessarily unique

Assume $\mathbb Q[\sqrt{d}]$ is a Euclidean Field and $\alpha$, $\beta$ are two quadratic integers in $\mathbb Q[\sqrt{d}]$, so that there exists integers $\gamma$ and $\delta$ in $\mathbb ...
2
votes
2answers
41 views

How to smoothly approximate a sign function

I have a function that defined as following $$f(x) = \begin{cases} 1, & \text{if $x > 0$ } \\ 0, & \text{if $x=0$ } \\ -1, & \text{if $x<0$ } \end{cases}$$ In practice, the $f(x)$ ...
1
vote
0answers
11 views

Apply the Euclidean algorithm to $2$ and $1 − 3i$ in the integers of $\mathbb Q[\sqrt {-1}]$

Question: I have to apply the Euclidean algorithm to $2$ and $1 − 3i$ in the integers of $\mathbb Q[\sqrt {-1}]$ My Solution: Since $N(2) = 4$ and $N(1–3i) = 10,$ we must start by dividing $1–3i$ ...
1
vote
1answer
17 views

Find GCD in Q[√3] assuming it is defined

How do I find the GCD of 24 and 49 in the integers of Q[√3], assuming that the GCD is defined?
6
votes
1answer
99 views

How to prove that $a=z^{p}$ for some $z \in \mathbb{Z_{+}}$?

Claim : If for a positive, composite integer $a$ and an odd prime $p$, such that $\gcd(a,p)=1$, we are given $$ a^{p^{n-2}(p-1)} \equiv 1 \pmod {p^n} \ \forall \ n \geq 2 \ \ ;\ ...
4
votes
1answer
102 views

Sums of three cubes in arithmetic progression equal to a cube $x^3+(x+y)^3+(x+2y)^3 = z^3$

Using exhaustive search, small positive and primitive integer solutions to, $$x^3+(x+y)^3+(x+2y)^3 = 3 x^3 + 9 x^2 y + 15 x y^2 + 9 y^3= z^3\tag1$$ are, $$x,y = 3,1,\quad x+y =2^2$$ $$x,y = ...
2
votes
1answer
51 views
+300

How does class field theory help us deduce the splitting of nonprincipal prime ideals?

I had a general question about the significance of global class field theory. One of the goals, as I understand, is to answer the following question: Given $L/K$ abelian, $g$ a divisor of $[L : ...
0
votes
0answers
32 views

What's the meaning of $d^{\times } a$?

In the lecture notes, the last line of on page 51, what is the meaning of $d^{\times } a$ in the integral? Thank you very much.
3
votes
1answer
48 views

Find all $c\in\mathbb Z^+$ for which $\exists a,b\in\mathbb Z^+, a\neq b$ with $a+c\mid ab$ and $b+c\mid ab$

Find all $c\in\mathbb Z^+$ for which $\exists a,b\in\mathbb Z^+, a\neq b$ with $\begin{cases}a+c\mid ab\\b+c\mid ab\end{cases}$ For those $c$, prove only finitely many $(a,b)$ exist. ...
2
votes
2answers
28 views

Is $\mathbb Z[\frac{1+\sqrt 5i}2]$ a ring of fractions of $\mathbb Z[\sqrt 5i]$?

Does there exist a multiplicative set $S\subset \mathbb Z[\sqrt 5i]$ such that $\mathbb Z[\frac{1+\sqrt 5i}2]\cong S^{-1}\mathbb Z[\sqrt 5i]$? Since the multiplicative structure of $\mathbb ...
0
votes
1answer
22 views

Real numbers modulo $1$.

In teaching material of my professor I read "where $x_1,x_2,...,x_m$ are distinct real numbers modulo $1$". What is the definition of numbers modulo $1$? Intuitively I would say that there exist a ...
0
votes
0answers
18 views

How to use mobius-inversion to solve this problem?

Currently, I'm trying to solve this problem using mobius-inversion. the function f(d) means the number of (i, j, k) equals d, and function g(d) means the numbers that satisfying: d | (i, j, k). Then ...
4
votes
0answers
73 views
+100

Compare $\sum_{k=1}^n \left\lfloor \frac{k}{\varphi}\right\rfloor$ …

Given two integer sequences $\displaystyle A_n=\sum_{k=1}^n \left\lfloor \frac{k}{\varphi}\right\rfloor$ $ B_n=\left\lfloor\dfrac{n^2}{2\varphi}\right\rfloor-\left\lfloor ...
3
votes
2answers
266 views

Do you know Legendre's conjecture ? Has it been proved?

Legendre's conjecture: proposed by Adrien-Marie Legendre, states that there is a prime number between $n^2$ and $(n + 1)^2$ for every positive integer $n$. Has it been proved?
1
vote
0answers
20 views

How do we know the representation for $n_2 \lt n_1$ has less bases?

Given that we know the following is true: $n = \sum_{j=0}^{s} c_j 3^j, c_s \ne 0, c_j \in \{-1,0,1\}$ We are to determine the least number of weights to weigh any integral number of pounds up to 80 ...
1
vote
0answers
53 views

An elliptic curve for the multigrade $\sum^8 a_n^k = \sum^8 b_n^k$ for $k=1,2,3,4,5,9$?

I. The first solution to, $$\sum^6_{n=1} a_n^9 =\sum^6_{n=1} b_n^9$$ $$13^9+18^9+23^9-5^9-10^9-15^9 = 9^9+21^9+22^9-1^9-13^9-14^9$$ was found in 1967 by computer search by Lander et al. It stood ...
1
vote
2answers
85 views

Why must $n$ be even if $2^n+1$ is prime? [duplicate]

This is a necessary step in a problem I am working on.
1
vote
2answers
28 views

Proving that a nonzero integer n has a unique representation

This is the first proof I've written. Can anyone give me advice? I don't know if its valid, or if there are ways to improve / other ways to do it: Prove that each nonzero integer may be uniquely ...
-1
votes
0answers
29 views

An inequality of arbitrary degree > 2 having an unique integer solution [on hold]

The surprising unicity of n in the inequality below is an inmediate application of Fermat Last Theorem (FLT); it is equivalent to FLT actually and to prove it would be extremely difficult without FLT ...
3
votes
1answer
22 views

How do I find the coordinate relationship between numbers on a number spiral?

For instance, considering the number spiral below, If I wanted to say where the number $10$ was in relation to the number $18$, I might say something like relationship $(18,10) = (4,-2)$ since it is ...
1
vote
1answer
28 views

Grid Problem Proof

I have a 2x2 grid square say, I can fit a shape like this: Such that there is one missing square. I can arrange this in any way so that the missing square can be located anywhere. I can do ...
2
votes
1answer
26 views

Why is $c+q$ still a primitive root modulo $q$?

Question: Let $p$ and $q$ be distinct odd prime numbers. By considering primitive roots, we need to show $\exists c\in\mathbb{Z}$ with the property that: $\bullet$ $c^n\equiv 1\pmod{p}$ whenever $n$ ...
2
votes
1answer
42 views

Numbers of the form $5 \cdot 2^{n}-1$ divisible by $3^k$ for large values of $k$

Let $n_k$ be the smallest integer such that $5 \cdot 2^{n_k}-1$ is divisble by $3^k$ where $k$ is a positive integer. Can one say something about the growth of $n_k$ with respect to $k$ ? Is it ...
0
votes
4answers
205 views

Curves triangular numbers.

Sometimes you have to deal with this equation: $X^2+aX+Y^2+bY=Z^2+cZ$ $a,b,c$ - integer coefficients. I wrote below - to start a particular solution of Diophantine equations. To do this, use the ...
0
votes
0answers
18 views

Does this 'alternating' series with $\Lambda(n)$ converge for all $\Re(s)>0$?

The following equation is well known and valid for $\Re(s)>1$: $$\log\big(\zeta(s)\big)=\sum_{n=2}^\infty \dfrac{\Lambda(n)}{\log(n)\,n^s}$$ where $\Lambda(n)$ is the Von Mangoldt function. Take ...
2
votes
1answer
47 views

How many solutions to $x^d\equiv a\pmod {p}$?

If $\gcd(d,p-1) = 1$, there is a unique solution to $x^d \equiv a \pmod p$. If $\gcd(d,p-1) > 1$, there are exactly $d$ solutions to $x^d\equiv a\pmod p$. $p$ prime, $d\ge 1$, ...
2
votes
1answer
20 views

Prove that powers of any fixed prime $p$ contain arbitrarily many consecutive equal digits.

Prove that powers of any fixed prime $p$ contain arbitrarily many consecutive equal digits. It is an intuitive re-statement of Baltic Way 2012 (I think there are shortlists in Baltic Way every ...
11
votes
2answers
242 views

Sum of 16 squares?

It is known that for n=1,2,4,8 the sum of squares is multiplicative. See for instance "Naive Lie theory" Stillwell for a good reference. It happens to be that these identities are the result of the ...
3
votes
3answers
36 views

Split 16 Consecutive Integers into Two Subsets of 8 Integers

Show that any given set of sixteen consecutive integers {$x+1,x+2,\ldots,x+16$} can be divided into two eight element subsets with the properties that they have the same sum, the sums of the squares ...