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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8 views

Difference between generalized cuban primes and cuban primes.

I have been studying cuban primes, and while the official definition of cuban primes contains only two variations, I have also seen a reference to generalized cuban primes, which has a much larger ...
0
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0answers
22 views

Every element in a finite field E is a sum of 2 squares.

I have a exercise of field. Prove that every element in a finite field E is a sum of 2 squares (if z = $a^2$ then we can write z = $a^2 + 0^2$. I have tried to use a quadratic residue but no result.
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1answer
21 views

$x^2 \equiv a$ mod p, $x^2 \equiv b$ mod p, and $x^2 \equiv ab$ mod p, prove either all three are solvable or exactly one

Let p be an odd prime and a, b ∈ Z with p doesn't divide a and a doesn't divide b. Prove that among the congruence's $x^2 \equiv a$ mod p, $x^2 \equiv b$ mod p, and $x^2 \equiv ab$ mod p, either all ...
-3
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2answers
26 views

Is $ n^{2} + 1 $ divisible by $ 7 $? By $ 13 $?

1) Does there exist a positive integer $ n $ such that $ n^{2} + 1 $ is divisible by $ 7 $? Prove assertions. 2) Does there exist a positive integer $ n $ such that $ n^{2} + 1 $ is divisible by $ 13 ...
1
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1answer
14 views

-a is also a quadratic residue mod p

Let p be an odd prime and let a be a quadratic residue modulo p. Prove that −a is also a quadratic residue modulo p if and only if p ≡ 1 mod 4.
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2answers
22 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
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2answers
21 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 ...
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0answers
15 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
16 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} = ...
0
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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} ...
2
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2answers
34 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 ...
0
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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 ...
-1
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1answer
30 views

Is $\pi(y)\pi(x+k)\ge\pi(x)\pi(y+k)$? [on hold]

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
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0answers
26 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
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0answers
17 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
2answers
34 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
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1answer
17 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 ...
0
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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 $z$.
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1answer
26 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
42 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
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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
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1answer
18 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?
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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.
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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 ...
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0answers
19 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 ...
2
votes
2answers
29 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 ...
1
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2answers
51 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 $x^2-Dy^2=r^2$). ...
1
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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 ...
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2answers
87 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.
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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 ...
3
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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 ...
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
43 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
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0answers
19 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
22 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 ...
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
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, $d\ge 1$, ...
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 ...
1
vote
1answer
29 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 ...
6
votes
1answer
64 views

Only finitely many $a, b$ such that $2+3^n+5^{n^2}=2^a7^b$ for some $n$?

Let $(a,b)$ be a pair of positive integers such that $$2+3^n+5^{n^2}=2^a7^b$$ for some positive integer $n$. Is it true that there are only finitely many such pairs? I don't know the answer to such ...
10
votes
2answers
79 views

If a number is of type $n^n$, how to identify that? Example: 256 is $4^4$, 3125 is $5^5$

If a number is of type $n^n$, how to identify that? Example: 256 is $4^4$, 3125 is $5^5$. I have to write a code for that.
5
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0answers
41 views

Summation of a function

Let $n$ is a positive integer. $n = p_1^{e_1}p_2^{e_2}...p_k^{e_k}$ is the complete prime factorization of $n$. Let me define a function $f(n)$ $f(n) = p_1^{c_1}p_2^{c_2}...p_k^{c_k}$ where $c_k = ...
1
vote
1answer
48 views

Solve $ 1^2+2^2+3^2+\cdots +k^2=n^2 $ in $ \mathbb{Z}^+ $ [on hold]

Find all $ (n,k)\in\mathbb{Z}^+ $ such that $ 1^2+2^2+3^2+\cdots +k^2=n^2 $
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votes
4answers
57 views

Let $p$ be an odd prime. Show that every prime divisor of $(2^p) - 1$ is greater than $p$. [on hold]

Let $p$ be an odd prime. Show that every prime divisor of $(2^p) -1$ is greater than $p$.
1
vote
1answer
53 views

Let $p$ be an odd prime, and let $n$ be an integer not divisible by $p-1$.

Let $p$ be an odd prime, and let $n$ be an integer not divisible by $p-1$. Show that $\sum x^n \equiv 0 \pmod{p}$, when the sum is over all $x$ with $0\le x\le p-1$. Some help with this practice ...
2
votes
2answers
35 views

Finding all integers satisfying an equation?

My task is to find all $x,y\in\mathbb{Z}$ which satisfy $$ 2x^2+2xy+y^2=25\quad\quad\text{(1)} $$ Is there a general approach of solving a task like this? If not, have you got tips for me, that might ...
2
votes
1answer
18 views

Explicit calculation of residue field in Cyclotomic integers

I would like to show that $(1-\zeta)$ is a prime ideal in $\mathbb{Z}[\zeta]$, where $\zeta=\zeta_p =e^{\frac{2\pi i}{p}}$, for a prime $p$. I am aware that we can show $(1-\zeta)=(1-\zeta^i)$, for ...
4
votes
0answers
65 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 ...
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 \ \ ;\ ...