Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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1
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1answer
69 views
+50

The intersection points are collinear

It is given a hexagon inscribed in a conic section. I want to prove that the pairs of opposite site intersect at three points of the projective plane that are collinear. How could we do this? ...
1
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1answer
20 views

The main involution on $ M_{2}(F) $ and it's extension to $ M_{2}(F_{\mathbb{A}}) $.

I'm presently reading through a paper of Shimura's; "Special Values of the Zeta Functions Associated with Hilbert Modular Forms". In the paper he defines $ \iota $ to be the main involution of $ ...
1
vote
1answer
138 views
+150

Flexes of cubic curve

Which are the flexes of the cubic curve of Fermat $$x^3+y^3+z^3=0$$ at $\mathbb{P}^2(\mathbb{C})$ ? Could you give me a hint how we could find the flexes? Do we have to use maybe the following ...
0
votes
1answer
26 views

Application of the Jacobian

I have been stuck on this question for a while now to no success. Help would be appreciated. Consider x and y such that (x, p) and (y, p) = 1. For what p does their exist x and y such that $x^2 + ...
0
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3answers
31 views

Chinese Remainder Theorem Finding the Modulo

Find numbers $t,u,v$ so that $33t+2 = 20u+13 = 29v-1 $ This is a Chinese Remainder Theory problem, but the problem I am having is finding what are the appropriate modulo. I figure it is easiest to ...
1
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2answers
69 views

Fermat's theorem, sum of prime squares.

By Fermat's theorem, a prime $p$, is a sum of two squares if and only if $p \equiv 1 \pmod 4$. I am wondering if there is any extension of this theorem or result that will give me the primes of the ...
2
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1answer
28 views

Do these properties characterize number rings?

Suppose $R$ is a Dedekind domain with the following properties: at every prime of $R$ the residue field is finite; fibers of the map $\text{Spec }R\to \text{Spec } \mathbb Z$ are finite. Is $R$ ...
10
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3answers
48 views

Is this sequence a recurrence relation?

$21, 36, 55, 60, 67, 68, 92, 93, 125$ I thought maybe it's $T_{2n + 4} + 2(n + 4)$ and I tried a few other formulas involving triangular numbers. I also tried variations on the Fibonacci sequence, ...
0
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1answer
13 views

For $m$ distinct fields among $\mathbb{Q}(\theta_1),\ldots,\mathbb{Q}(\theta_n)$ prove that $m\mid n$ and each field occurs $n/m$ times

I'm having some trouble with this problem, and I wanted to know if someone could help me out. Let $K=\mathbb{Q}(\theta)$ be an algebraic number field of degree $n$. Let ...
0
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2answers
24 views

How to prove $x^2=-1$ has a solution in $\mathbb{Q}_p$ iff $p=1\mod 4$

Let $p$ be prime and let $\mathbb{Q}_p$ denote the field of $p$-adic numbers. Is there an elementary way to prove $x^2=-1$ has a solution in $\mathbb{Q}_p$ iff $p=1\mod 4$? I need this result, but I ...
4
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6answers
203 views

How can I prove the pattern $\sqrt{1 + 155555…5} = 2 \sqrt{3888…89}?$

How can I prove this $$\sqrt{1+155}=2\sqrt{39}$$ $$\sqrt{1+1555}=2\sqrt{389}$$ $$\sqrt{1+15555}=2\sqrt{3889}$$ $$\sqrt{1+155555}=2\sqrt{38889}$$
2
votes
1answer
18 views

Proof with exact sequence of modules

I'm trying to prove that if the sequence $$ M \xrightarrow{\varphi} W \rightarrow 0$$ is exact with $ W $ being a free module, then $ M \simeq \ker{\varphi} \oplus W $ What I got is that since $ W ...
9
votes
1answer
174 views

Gap in count of bases for which a number's representation is palindromic (by concatenation of decimal representations)

I have written a small program that calculates the base representation of a number, and checks if that representation is a palindrome (also check if the length is at least 2, since a length of 1 is ...
46
votes
4answers
4k views

Is 128 the only multi-digit power of 2 such that each of its digits is also a power of 2?

The number $128$ can be written as $2^n$ with integer $n$, and so can its every individual digit. Is this the only number with this property, apart from the one-digit numbers $1$, $2$, $4$ and $8$? ...
0
votes
0answers
16 views

What is a nice way to call continuants?

I'm reading this paper : http://www.numbertheory.org/pdfs/continuant.pdf and here is a definition for continuant : http://en.wikipedia.org/wiki/Continuant_(mathematics) Let $\{a_n\}$ be a ...
4
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7answers
83 views

Primality of number 1

Is number 1 prime as per the definition of prime numbers? Because as per the definition for being prime it should be divided only by 1 and number itself.
3
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0answers
48 views

Comparing up-arrow's

Is it true that $$3\uparrow^{n+1} 3\ >\ n\uparrow^n n $$ holds for every $n\ge 1$ Since $3\uparrow^{n+1}3=3\uparrow ^n 3\uparrow ^n 3$ and $3\uparrow^n3$ is much bigger than $n$ for $n\ge 3$, ...
0
votes
2answers
67 views

Make a prime number from specified number, by concatenating some more digits on its right?

I am given a number, I don't know whether it's prime or not. The algo says, For eg - Step 1 - Convert char to ints. (Hello - 72101108108111) Ascii values Step 2 - Make a large number. Convert char ...
1
vote
1answer
22 views

If (an)→ L, an > 0 for all n ∈ N, and L > 0, then prove that √an → √L .

To be honest, I don't even understand what the question is asking, and have no idea how to answer it. Any guidance would be great. I know convergent/divergent definitions, as well as basic limit laws, ...
4
votes
1answer
56 views

Show that $\limsup \pi(n)/n = 0$ with elementary techniques.

Suppose $S$ is a set $S \subseteq N$ and suppose $$\lim_{n \to \infty} \frac{|Z_n \cap S|}{n} = c \in (0,1).$$ How do we prove, using elementary means, that there is a composite number in $S$? If ...
1
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3answers
47 views

Prove that the sequence $\cos(n\pi/3)$ does not converge

EDIT: Using a rigorous formal proof, I need to prove that this sequence does not diverge. I of course understand why it doesn't converge... $n=1$ to infinity of course. So, I have a bit of trouble ...
0
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1answer
20 views

Deriving Thue's lemma from Minkowski's convex body theorem

I'm trying to find an alternative proof of Thue's lemma, stating that for $ p \in \mathbb{N}, a \in \mathbb{Z}_p^* $ $$ \text{The congruence } x \equiv ay \pmod{p} \text{ has a non-zero solution such ...
0
votes
0answers
25 views

What is the domain of continued fraction?

I'm trying to formally define (generalized) continued fraction. Consider $[i;\sqrt2,i,i]$. This is not well defined since $i+\frac{1}{i}=0$. What would be a domain of continued fraction? (As a ...
0
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0answers
25 views

A question on polylogarithms and sums like $Li_{3}(z) = \displaystyle\sum_{n=1}^{\infty}\frac{1}{n^3} = \zeta(3)$.

I have been working with polylogarithms $Li_{n}(z)$ and have a question concerning them. Things like $Li_{2}(z)$ and $Li_{4}(z)$ can be computed and have an exact solution. For example $Li_{2}(z) = ...
1
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1answer
27 views

Why are special numbers important? (Such as fermat prime, mersenne prime)

Whenever I studied topics in mathematics, I found those topics are important in purely mathematical sense and I could see some motivations. However, I cannot see neither motivation nor importance of ...
2
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1answer
41 views

Well ordering principle and prime factoriation

Is it possible to prove the uniqueness of prime factorisation of natural numbers by the well ordering principle ? My attempt : Let S be the set of all natural numbers whose prime factorisation is ...
3
votes
1answer
49 views

How find prime numbers $p_{i}$ such $p_{1}+p_{2},p_{2}+p_{3},p_{3}+p_{4},\cdots,p_{n}+p_{1}$ is square number

Question: Let $n\ge 5$ be an odd number, show that: there exist (or does not exist) primes $p_{i}\:;\:i=1,2,\cdots,n$ such that $$p_{1}+p_{2},p_{2}+p_{3},p_{3}+p_{4},\cdots,p_{n}+p_{1}$$ all ...
0
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0answers
36 views

Modular equation with modulo 9.

$$a^3 \equiv x \mod 9 \iff x \in \{ -1,0,1\}$$ Why it is true? I don't understand. Thanks in advance for everyone.
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0answers
34 views

Reasoning behind the approach of finding modular multiplicative inverse

We know that if we want to find out the value of $\frac ab\pmod m $ we have to find out the modular inverse of b. If the modular inverse of b is x ,then we multiply x with a. Now ...
0
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1answer
28 views

the largest chain of number

I found this problem on the internet: What is the largest chain of numbers that complies with that every number in the chain/list is a divisor divisor in the next number. For example 1 - 6 - 18 ...
2
votes
0answers
30 views

divisibility and k-power sum

Let $a_{1},\dots,a_{n},\,n>2$ distinct natural numbers. Prove that if $p_{1},\dots,p_{r}$ are prime numbers and they divide $a_{1}+\dots+a_{n}$ then exists an integer $k>1$ and a prime $p\neq ...
0
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0answers
34 views

Growth rate of $f(f(n))$, where $f(n)$ is the ackermann-function.

Let $$f(n)\ :=\ n \uparrow^n n$$ and $$g(n)\ :=f(f(n))\ =\ f(n)\uparrow ^{f(n)} f(n)=n\uparrow^n n \uparrow^ {n \uparrow ^n n} n\uparrow ^n n$$ So, $g(n)$ is $f(n)$ applied twice. What is the ...
2
votes
0answers
35 views

prove two sets have the same g.c.d.

$a_n,b_n$ are two sequence valued in $[0,1]$ and $a_0=1,b_0=0 $. the following equation holds: $$a_n=\sum_{k=1}^{n}b_ka_{n-k}\tag{1}$$ $$A=\{n:a_n>0\}-\{0\}$$$$B=\{n:b_n>0\}$$ further ...
1
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0answers
64 views

Eigenforms for $\mathcal{S}_2(\Gamma_0(88))$

I'm having trouble with the following exercise (5.8.3) from Diamond and Shurman's text on modular forms (this isn't homework for class, I just wanted to work this out on my own): ...
0
votes
1answer
23 views

An indeterminate equation

Question: Calculate the indeterminate equation $x^2+y^2=z^4$, which satisfy $(x,y)=1,2|x$ and x,y,z are all positive integer. $(x,y)$ represents the greatest common divisor of x and y.Below is my ...
7
votes
4answers
118 views

Binary operation commutative, associative, and distributive over multiplication

Is there any binary operation that is commutative, associative, and distributive over multiplication? I asked this question in my head a while ago, and I posted it in various forums. However, having ...
1
vote
1answer
45 views

Name of Legendre symbol?

This may seem stupid question, but I'm curious about this. Generally, $(a/p)$ is called "the Legendre symbol" where $p$ is an odd prime, but I don't like this naming since this naming is not formal. ...
1
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0answers
23 views

Proving Euler’s congruence and Legendre

So the question is "Prove Euler's congruence $a^{\frac{p-1}{2}} \equiv \left(\frac{a}{p}\right) \bmod p$ for odd primes p and a in $\\Z$." So I know that $$\left(\frac{a}{p}\right) = \begin{cases} ...
3
votes
1answer
62 views

Calculating of genus of a curve

Let $C$ be a curve over $\mathbb{F}_q$ in projective plane. So $C$ can be done as zeroes of some gomogeneous polynomial $\in \mathbb{F}_q[x,y,z]$ with degree $n$. Whether is there algorithm which is ...
3
votes
1answer
78 views

Dirichlet density

How to solve the following exercise: Let $q$ be prime. Show that the set of primes p for which $p \equiv 1\pmod q$ and $$2^{(p-1)/q} \equiv 1 \pmod p$$ has Dirichlet density ...
0
votes
1answer
35 views

Prove that there are infinity many numbers you can't write in the form $a^{T(a)}+b^{T(b)}$.

Prove that there are infinity many numbers you can't write in the form $a^{T(a)}+b^{T(b)}$ where a and b are positive integers. T(a) represents the number of divisors number a has.
0
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2answers
81 views

Find all real real functions that satisfy the following eqation $f(x^2)+f(2y^2)=[f(x+y)+f(y)][f(x-y)+f(y)]$

Find all real functions $f:\Bbb R\rightarrow\Bbb R$ so that $f(x^2)+f(2y^2)=[f(x+y)+f(y)][f(x-y)+f(y)]$, for all real numbers $x$ and $y$. $f(x)=x^2$ is the only solution I think. So far I have got: ...
2
votes
0answers
33 views

Question on Dirichlet density

I did not understand the highlighted sentence of the exercise below: My question is: how does it follow that $f(x)=0$ has a solution mod $p$ implies that $f(x)$ (mod $p$) splits as the product of ...
-1
votes
0answers
47 views

Find all positive integers $n$ such that …

Find all positive integers $n$ for which $3^{2n}+3n^2+7$ is an perfect square. I got that $n$ has to be even. But then I didn't get anything.
11
votes
1answer
199 views

Solving $n^5+n^4-3=x^2\pmod p$

Prove that for every odd prime number $p$ there is a natural number $n$ such that the equation $n^5+n^4-3=x^2\pmod p$ has no solutions. So we have to understand that for each $p$ we can find $n$ ...
0
votes
1answer
43 views

Integers that are equal to the sum of their digits each raised to that digit's power [closed]

The number 3,435 has the interesting property where $$3435 = 3^3 + 4^4 + 3^3 + 5^5$$ That is, it equals the sum of its digits each raised to that digit's power. What is the next larger number with ...
0
votes
0answers
42 views

additive number theory: sums and products of subsets of integers

Suppose that $A$, $B$ are finite subsets of the integers. Consider the subset $E$ of $A+B$ consisting of all elements $s$ of $A+B$ that can be written uniquely as $s=a+b$, where $a\in A$ and $b\in B$, ...
8
votes
8answers
1k views

“Integer average” of two integer numbers

Suppose two arbitrary integer numbers $a$ and $b$. I'm looking for some function $f(a,b)$ with the following properties: $f(a,b)\in\mathbb{Z}$. $f(a,a)=a$. $f(a,b)=f(b,a)$. $\min\{a,b\}< ...
4
votes
1answer
62 views

Fairly good semiprime estimate

I have found a nice estimate for the semiprime counting function \begin{align} &f_{2}(x):=x \log \left( \log (x)/\log \left( a+a/ \exp\left( (\log (\log (x)-2)-1)^2/2\right) (\log (x)-2) \right) ...
1
vote
0answers
18 views

Specific question on dirichlet density

In a notes I found the following exercise and solution: I have a question. In the proof I admit the statement "the Dirichlet density of these prime ideals is $1/2$ " but i do not understand why the ...