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

learn more… | top users | synonyms (1)

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
175 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
5k 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
votes
7answers
84 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
votes
0answers
52 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
vote
3answers
49 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
votes
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
votes
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
vote
1answer
28 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
votes
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
51 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
votes
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.
0
votes
0answers
35 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
votes
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
votes
0answers
38 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 ...
2
votes
0answers
68 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
119 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
vote
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
79 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 ...
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
200 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 ...
0
votes
0answers
36 views

any online video course for analytic number theory or elementary number theory?

All: I am looking for online video course on analytic number theory for self-study. On Youtube, there are a few seminars, but no complete course for a semester or a year. Can anyone point out if ...
0
votes
1answer
20 views

What is the name of this function? index?

Burton- Number theory p.163 Let $n\in\mathbb{Z}^+$. Let $r$ be a primitive root mod $n$, so that $<r>=\mathbb{Z}_n^*$ Let $a\in \mathbb{Z}_n^*$ Let $k$ be the smallest ...
5
votes
3answers
89 views

A good introductory book on Ring and Field theory with a view towards Number Theory ?

Please suggest some good introductory books on Rings&Fields with a view towards Number Theory ?
1
vote
4answers
41 views

Upper limit for the Divisor function

For a a ponsitive inreger $n$, let $d(n)$ denote the number of divisors of $n$. I'm trying to prove that: 1) For $n>6$ we have $d(n)\leq\frac{n}{2}$; 2) For $n>12$ we have ...
0
votes
0answers
27 views

Mobius Function and Liouvlle's Function

$\lambda(n)$= $\sum_{d^2|n}$ $\mu(n/d)^2$ and $\mu^2(n)$= $\sum_{d^2|n}$ $\mu(d)$ Having a little bit of trouble here.Can I use the fact that $\sum_{d|n}\lambda(n)$ is a characteristic function for ...
7
votes
3answers
133 views

Prove that there are infinitely many $m,n$ for which $\frac{m+1}{n}+\frac{n+1}{m}$ is an integer

Prove that there are infinitely many pairs of positive integers (m,n) such that $\frac{m+1}{n}+\frac{n+1}{m}$ is a positive integer. I tried following: clearly $(1,1)$ satisfies the condition. ...
1
vote
1answer
25 views

is Legendre Symbol related with inverse?

I want to show that If $p$ is an odd prime, then $\sum_{ j = 1 }^{ p-2 } \left(\frac{j^2 + j}p \right) = -1$ (where $\left(\frac\cdot\cdot\right)$ is Legendre Symbol). My teacher said I should ...
5
votes
2answers
59 views

Prove or disprove that $p_n > e^{p_n - p_{n-1}}$ for large enough $n$.

Let $p_n$ denote the $n$-th prime. Prove or disprove that for large enough $n$ we have $$p_n > e^{p_n - p_{n-1}}.$$ The inequality trivially holds for all the twin primes larger than $7$. With ...
3
votes
1answer
28 views

Modular arithmetic with huge modulus?

When the dividend is some huge power but the modulus is not so big, I can use modular exponentiation. But how can I compute the residue when the modulus is, for example, $2^{107} - 1$, a Mersenne ...
3
votes
0answers
71 views

Group generated by two polynomials

The $Homeo(\mathbb{R})$ be the group of all the homeomorphisms on $\mathbb{R}$, with the group operation of composition. Let $f(x)=x^3+\alpha$ and $g(x)=x^3+\beta$ be two elements of ...
2
votes
0answers
47 views

Finding newforms with Sage

I am new to Sage and modular forms. I have some conceptual questions. When I write sage: S = CuspForms(Gamma0(55),2,prec=14) sage: S.new_subspace().basis() ...
0
votes
0answers
26 views

ALGORITHM Multiplication of Integers from “discrete math and its applications 7th edition ” book

Please can you help me to understand the "italic text" How many additions of bits and shifts of bits are used to multiply a and b using Algorithm 3"see the the attached photo"? Solution: ...
6
votes
1answer
165 views

How can I know if $2^{2^{2^{2^{2}}}}+1=?$ is prime?

I could calculate the following prime numbers $$2+1=3$$ $$2^{2}+1=5$$ $$2^{2^{2}}+1=17$$ $$2^{2^{2^{2}}}+1=65537$$ Are the following numbers prime??? $$2^{2^{2^{2^{2}}}}+1=?$$ ...