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)

3
votes
0answers
58 views

Properties preserved under the “reversal” of a recurrence equation

Consider the recurrence equation $u_n = f(u_{n-1},\ldots, u_{n-k})\;,$ defined for $n=0,1,2\ldots$. If this is a linear recurrence and the coefficient function on $u_{n-k}$ is nonzero in ...
3
votes
0answers
192 views

Approximation of a real number as a linear combination of two reals with coprime integral coefficients

Given two nonzero real numbers $x$ and $y$ such that $y/x$ is irrational, a real number $z$ to be approximated, and a tolerance $\epsilon$, give me an algorithm that will provide coprime integers $a$ ...
3
votes
0answers
174 views

Hauptmoduls for modular curves

If I have a modular curve, how does one in general find a Hauptmodul for this curve?
3
votes
0answers
292 views

Rational independence

The specific question is the following: I am given a set of $[L/2]$ numbers $$g(n) = \sqrt{ c(n)^2 + \alpha c(n) + \beta},$$ where $c(n) = \cos(2\pi n/L)$ (so both $c(n)$ and $g(n)$ depend on $L$ ...
3
votes
0answers
195 views

The discrete Fourier transform of a Dirichlet charachter

I usually work in number theory so I am not familiar with Fourier transforms, I have read up on them and know the basics but it never seems to be in number theory language. I am trying to find the ...
3
votes
0answers
139 views

How to show that if K contains a complex root of unity, then every nonzero element of K has positive norm?

I am trying to prove the following assertion: If an algebraic number field $K$ contains a complex root of unity, then the norm of every nonzero element of $K$ is positive. I think this is supposed ...
3
votes
0answers
250 views

Counting Solutions of Diophantine Inequalities

I understand that Diophantine Analysis is an enormous field! Without first determining the solution set, suppose I'd like to calculate the number of non-negative integer solutions $(x,y,z)$ of ...
2
votes
0answers
34 views

Cramér's Model - “The Prime Numbers and Their Distribution” - Part 2

Following a previous question (here you'll find an introduction): The book states that using the convergence of the binomial distribution towards the Poisson distribution, it's easy to show that ...
2
votes
0answers
36 views

What's the proof for the #integers less than $n$ that can be expressed as the sum of two squares is $\frac n{\sqrt{\log n}}$?

This result is used in the Erdos' Distance problem, in the Landau-Ramanujan constant, but I can't find a proof anywhere. http://en.wikipedia.org/wiki/Erd%C5%91s_distinct_distances_problem ...
2
votes
0answers
35 views

Bernoulli Conjecture on $B_{2^n}$

So in a recent question I was trying to prove that $2^n-1$ will never be a Carmichael number (Can a Mersenne number ever be a Carmichael number?), I was going to prove it true as long as a certain ...
2
votes
0answers
29 views

Fermat pseudo primes

Is it possible for a number of the form $2^p-1$ with $p\in \mathbb{P}$ (the primes) to satisfy $3^{2^p-2}\equiv 1\pmod {2^p-1}$ and not be a prime? In other words, can a Mersenne number be a Fermat ...
2
votes
0answers
88 views

“Remainder” operation in mod 2^32

I debated posting this here, in the cryptography SE, or the programming SE. Obviously, I chose here, but I'm not confident in my choice... I'm attempting to "undo" a function, but I've hit a slight ...
2
votes
0answers
27 views

Polynomial Density Theorem

So I was consider Lagrange's 4-square theorem and came up with this generalization: Given a polynomial with rational coefficients $$P(x) = a_0 + a_1x + ... + a_nx^n$$ Determine if there exists ...
2
votes
0answers
134 views

Successive ratios of a sequence, is this limit true?

The natural numbers $1,2,3,4,5....$ can be calculated as the row sums of the triangle $T(n,k)$ equal to $1$ if $n \geq k$ and $0$ otherwise: $$\displaystyle T = \left(\begin{matrix} ...
2
votes
0answers
46 views

Integer Solutions To Linear Equation

$$a*q_1+b*q_2=c$$ $$a*q_3+b*q_4=f$$ $q_1, q_2, q_3, q_4$ rational numbers, $c,f$ integer Given $q_1, q_2$ can you construct all solutions $(a,b)$ where $c,f$ is intenger I made an edit since the ...
2
votes
0answers
31 views

Approximating real vectors in the unit hypercube by rational vectors in the unit hypercube

I am currently looking for an error bound for an problem where I have to approximate a real vector in the unit hypercube using a rational vector in the unit hypercube. Specifically: given a ...
2
votes
0answers
29 views

Intuition behind Khinchin's constant

Khinchin proved that For almost all reals $r$ with continued fraction representation $[a_o; a_1, a_2, \dots ]$ the sequence $K_n = \left(\prod_{i=1}^{n} a_i\right)^{1/n}$ converges to a ...
2
votes
0answers
50 views

Prove $a^m\equiv a^{m-\phi(m)}\pmod m$ for all positive integers

Prove that if $a,m$ are positive integers, then $$a^m\equiv a^{m-\phi(m)}\pmod m.\tag 1$$ If gcd$(a,m)=1$ then this is Euler's theorem. Denote gcd$(a,m)=k$ and $a=xk,m=yk$ then we need to prove ...
2
votes
0answers
26 views

Semiprime asymptotic step function

Since $$\pi_{(2)}(x)=\sum_{i=1}^{\pi(x^{1/2})}\left(\pi\left(\dfrac{x}{\text{p}_i}\right)-i+1\right),$$ where $\pi_{(2)}(x)$ denotes the semiprimes and $\text{P}_i$ is the $i$th prime, an asymptotic ...
2
votes
0answers
52 views

totally split primes in a number field

I have to show: For any number field $K$, there are infinitely many prime numbers $p \in \mathbb{N}$, that are totally split in $K$. I think have already shown (with some hints my professor gave) ...
2
votes
0answers
31 views

A generalisation of Roth's result on Diophantine approximation?

It is a celebrated result of Roth that algebraic numbers cannot be approximated by rationals too accurately: $\newcommand{\norm}[1]{\left\lVert #1 \right\rVert_{\mathbb{R}/\mathbb{T}}}$ Theorem ...
2
votes
0answers
27 views

Meaning of tamely ramified extension.

Let $K$ be a complete field with respect to a discrete nonarchimedean valutaion. We denote $A$ and $\mathfrak{p}$ as its valuation ring and valuation ideal, respectively. For a finite Galois extension ...
2
votes
0answers
70 views

Points in a general Cantor set

We often look at the Cantor set with the construction that keeps removing the middle thirds of the remaining line segments at each iteration. Corresponding to this construction, we can determine ...
2
votes
0answers
58 views

Birthday problem & primes

Let $\pi_k(n)$ be the almost prime counting function, then $\pi_k(2^kn)$ reaches a max value, since $\pi_k(2^kn)=\pi_{k+1}(2^{k+1}n)$ for large enough $k$. (eg, ...
2
votes
0answers
82 views

Questions about central polygonal numbers $1, 2, 4, 7, 11, 16, 22, 29, 37, 46,\cdots$

Formula for Central polygonal numbers is $\frac{n(n+1)}{2} + 1$, if $n=1$ or $n$ is prime, we get the new sequence $A$: 2, 4, 7, 16, 29, 67, 92, 154, 191, ... It seems that all primes either is ...
2
votes
0answers
40 views

Tau Summatory Function

It is well known that the divisor summatory function can be calculated in $O(x^{1/2})$ via $$D(x)=\sum_{n\le x} d(n) = 2 \sum_{k=1}^{\lfloor \sqrt{x}\rfloor} \lfloor\frac{x}{k}\rfloor - \lfloor ...
2
votes
0answers
67 views

Unexplanied pattern from increasing rational sequences

I've stummbled upon some strange pattern when working with series of rational numbers. If anyone could shed light on this phenomenon I will be most greatful. Backgroud: Working in an integer lattice ...
2
votes
0answers
208 views

Count swap permutations

Given an array A = [1, 2, 3, ..., n]: ...
2
votes
0answers
66 views

Find all pair(s) of positive integer $(a,b)$ such that $\frac{a^2}{2ab^2 -b^3+1}$ is also positive integer too?

Another number theory problem. I can find the small value of $b$ such that 0,1,2. But, I cannot find the upper limit of $b$, such that the value of $b$ is limited. How can I find the solution ...
2
votes
0answers
39 views

A better Reference than Andre Weil's Basic Number Theory

I want to get a feel for Adeles. I have been suggested to read the first 4 chapters of Andre Weil's Basic Number Theory. I am very confused by the writing style and conventions (like a field need not ...
2
votes
0answers
69 views

arithmetic sequence $8n+1$ and the collatz conjecture

Is it a known result that if for all $n$ the collatz sequence of $8n+1$ lead to $1$, all natural numbers will?
2
votes
0answers
20 views

Finding coefficients of the min polynomial of an $n\times n$

Given an $n\times n$ matrix, for ease assume this matrix is over the $F_m$. What we know about min poly is the the non-zero components of the min polynomial for this case, ie if there is $x^2$, or ...
2
votes
0answers
30 views

binomial coefficents and composite numbers

Given a binomial coefficient $${n \choose k}$$ With $n>3$ and $k\neq 1$ or $n-1$, is the binomial cofficient always a composite number?
2
votes
0answers
31 views

Growth rate of arithmetical function

I'm interested in how one would estimate the growth rate of $$f(n)=\sum_{k\le n}\mu^2(k)\log(k)$$ I.e. sum of logarithms of square free integers. I can think of some trivial methods in my head ...
2
votes
0answers
59 views

Is this a conjecture or an already existing one??

Does the following inequality hold? $p(n)\leq 2^n,$ where $p(n)$ is the $n$th prime. If this is true then it follows that: If $p(n)=p(m)^x+p(o)^y$, then $\max[x,y] \le n$.
2
votes
0answers
49 views

Numbers for $k^{th}$ power of sum of digits equal the sum of digits of $k^{th}$ power

Let us denote $D_*(n)$ to be the sum of digits of $n$. Then we can find numbers satisfying the formula $$D_*(n^2)=D^2_*(n)$$ Like : $11^2 =121 ,\; \;(1+1)^2= 1+2+1$ , other like $12^2=144, \; \; ...
2
votes
0answers
75 views

Sum of $k$th power of first $n$ natural number (power sum)

I was working on a problem which involves computation power sum (summation of $k^{th}$ power of first natural number), can someone help me how to simplify the below equation. I can compute power sum ...
2
votes
0answers
61 views

Prove the inequality for composite numbers

Is it true that $c_m+c_n$ $>$ $c_{m+n}$ for all $m$, $n$ $\in$ $\mathbb{N}$? Though the result seems true, I can't get a solution. Even the bounds on $c_n$ obtained from Prime Number Theorem ...
2
votes
0answers
60 views

Formula for sequence of integers

I am trying to compute the Taylor series of $f(x) = \sqrt{-\ln(x)}$. I compute the derivatives of $f(x)$ and evaluate them in the point $x=1/e$. The resulting expressions have the following ...
2
votes
0answers
22 views

Asymptotics of the differences between successive zeta zeros

Does anyone know what the asymptotic of the differences between successive zeta zeros is? Update It appears that $\zeta(n)$ is not a bad asymptotic, when the data range is stretched: ...
2
votes
0answers
33 views

Subfactorial primes

So I just did some stuff and from what I can see, if y > x then !x + !y can only be prime if y = x+1 (apart from a few small exceptions near the start of the list. I don't know anything about ...
2
votes
0answers
60 views

All those unit fractions add to 1?

Consider $$S(n)=\{x \mid x=(a_1 ,a_2,a_3 \cdots a_n) \text{ where } \sum_{r=1}^{n}\frac{1}{a_r} =1 \}$$ Now let $|S(n)|$ denote the cardinaly (order) of set $S(n)$. Thus: $S(1)= \{(1)\} \implies ...
2
votes
0answers
44 views

$n$th prime bounded from above?

Let $p_n$ be the $n$th prime, $p_n\#\equiv\prod_{k=1}^{n}p_k$ (primorial), and $\sigma(n)=\sum_{d|n}^{}d$ (divisor function). Does $\text{exp}\bigg(\dfrac{\pi^2}{6 ...
2
votes
0answers
30 views

Inverse logarithmic integral

If the expansion of the logarithmic interval is$$\text{li}(n) = \log \log n + \gamma + \sum_{k=1}^\infty \dfrac{(\log n)^k}{k! k}$$ what is the inverse of the function?
2
votes
0answers
31 views

Why is the additive identity different to the multiplicative and exponential?

The additive identity is 0. The multiplicative and exponential identities are 1. Exponentiation is repeated multiplication, and multiplication is repeated addition. If you go up a further level, to ...
2
votes
0answers
30 views

Why is the Legendre symbol $(\frac a p)$ not defined if $p | a$ in some books?

Why is the Legendre symbol $(\frac a p)$ not defined if $p | a$ in some books ? In some textbooks I've come to notice that the legendre symbol $(\frac a p)$ is not defined if $p | a$, where $p$ ...
2
votes
0answers
59 views

Consequence of the Riemann Hypothesis

So I watched this video: http://m.youtube.com/watch?v=rGo2hsoJSbo And it included the fact that a consequence of RH is that there will always be a prime number between consecutive cubic numbers. I ...
2
votes
0answers
21 views

System of equations - modular arithmetic

I am asked to solve the following..... Let $n\in \mathbb{N}$ and suppose that $a,b,c,d,k,l\in\mathbb{Z}$. Consider the system $ax + by \equiv k$ mod $n$ and $cx+dy \equiv l$ mod $n$. Let $D=ad-bc$. ...
2
votes
0answers
27 views

Which integers are a product of partition numbers?

What can be said about the set of positive integers representable as a product of the form $p(n_1)\cdots p(n_r)$ for the partition function $p(n)$ ? Such numbers $k$ arise as the number of distinct ...
2
votes
0answers
73 views

Major arcs in the proof that every odd number is the sum of at most 5 primes

In his proof that all odd numbers greater than 1 are the sum of at most 5 primes, Terence Tao uses one large major arc around 0 rather than small ones around the rationals, which I am more accustomed ...