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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0
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0answers
14 views

Semiprime error margin

As an extension to this question, the plot below shows $$\pi(x)-R(x)\ \ \text{ (blue)},$$ $$\pi_{(2)}(x)-smoothed\left[ ...
0
votes
1answer
22 views

Multiplicative functions and Chinese remainder theorem

$ p $ is a nonconstant polynomial with integer coefficients.Define the function $\chi_p(n)$ as the number of zeros of $ p $ in $\mathbb{Z}_n$ for $ n > 1 $, and $ \chi_p(1) = 1 $. e.g., consider $ ...
2
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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 ...
5
votes
1answer
99 views

Units of $\mathbb{Z}[\sqrt[4]2]$

How would one compute the units in $\mathbb{Z}[\sqrt[4]2]$? According to one source, it can be shown that the fundamental units are $1 + \sqrt[4]2$ and $1 + \sqrt{2}$, but it does not specify the ...
0
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0answers
21 views

How to find a certain uppper bound (see details)?

What would be the most efficient way to find this upper bound? Given natural number n and a natural number d < n, find the ...
10
votes
1answer
354 views

Seeking proof for the formula relating Pi with its convergents

Could anyone try to prove that the below conjectured formula is valid for relating $\pi$ with ALL of its convergents - those, which are described in OEIS via A002485(n)/A002486(n) ? $$(-1)^n\cdot(\pi ...
3
votes
1answer
70 views

Probability that two random integers have only one prime factor in common

The probability that two integers picked at random are relatively prime is known to be $1/\zeta{(2)}=6/\pi^2\approx0.607927...$. Generalizing, the probability that $n$ random integers have $\gcd=1$ is ...
7
votes
0answers
107 views

What transcendental numbers are produced by $\sin{\alpha}$ when $\alpha$ is algebraic/constructible/rational (in radians)?

I know that by Lindemann–Weierstrass theorem(LW) sine and cosine of non-zero algebraic numbers (in radians) produce results that are transcendental. My question is what are the transcendentals ...
1
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0answers
20 views

How do I prove this equality involving ceilings and max?

For all $T \in \mathbb{N}$ the following holds, with $k \in \mathbb{Z}$ and $m, n \in \mathbb{N}$: $$\left \lceil \frac{k \cdot m}{n} \right \rceil + T - 1 = \max_{0 \leq i < T} \left \{ i + T ...
6
votes
2answers
130 views

Asymptotic density of powers of primes

I'm supposed to compute the asymptotic density of the set \begin{equation} \Pi(x):=\#\{p^k \leq x :p \;prime, k \in \mathbb{N}\} \end{equation} of prime powers less or equal to $x$, that is, compute ...
9
votes
2answers
136 views

How to prove that the number $1+4a_{n}a_{n+1}$ is a perfect square.

A sequence of integer $\{a_{n}\}$ is given by the conditions $a_{1}=1, a_{2}=12,a_{3}=20$,and $$a_{n+3}=2a_{n+2}+2a_{n+1}-a_{n}$$ show that for every postive integer $n$, the number ...
4
votes
3answers
102 views

Elementary, direct proof of when $5$ is a quadratic residue mod $p$

$\newcommand{\kron}[2]{\left( \frac{#1}{#2} \right)}$ It's easy to use Quadratic Reciprocity to show that $\kron{5}{p} = \kron{p}{5} = 1$ when $p \equiv \pm 1 \pmod 5$, and is $-1$ when $p \equiv \pm ...
0
votes
1answer
153 views

Find sum of all permutations

We call two arrays A and B with length n almost equal if for every i (1 <= i <= n) ...
-4
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2answers
78 views

Determine if the following functions are multiplicative… [closed]

Determine if the following functions are multiplicative $ a) \ f(n) = \gcd(n, k)$ for a fixed $k$ $\\b) \ \Lambda(p^e) = \ln p$ if p is a prime and $e\ge 1$, otherwise $\Lambda(n) = 0$. ...
-1
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0answers
48 views

To find the polynomial.

On adjacent forum hate formula. But the question is interesting and would like to have it clear. Theme there: ...
2
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1answer
35 views

Find the remainder [closed]

I need to find the remainder when $(59^{73})^{5!}$ Is divided by $37$. It has some binomial expression as well. I am not able to compute the powers.Please help Regards
3
votes
3answers
42 views

Finding domain of $f\text{ o }g$

I am having a small question, please don't close this before answering, I just want to know whether its a matter of convention or not. If $f(x) = \dfrac{1}{x}$ and $g(x) = \dfrac{1}{x}$ $ $ Then ...
8
votes
0answers
66 views

A sequence that avoids both arithmetic and geometric progressions

Sequences that avoid arithmetic progressions have been studied, e.g., "Sequences Containing No 3-Term Arithmetic Progressions," Janusz Dybizbański, 2012, journal link. I started to explore sequences ...
0
votes
1answer
27 views

Is there a quick way to obtain $a,b$ in $ax+by = z$ where $x,y,z$ are fixed and $x+1 = y$?

Suppose that all numbers are postive integers. Let $x,y,z$ be fixed/given and $x+1=y$. Then would there be a quick way to find set of solutions $(a,b)$ that satisfy $ax+by=z$? "Quick" would be ...
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0answers
93 views

Arithmetic progression and average of two prime numbers

Let $A=(a_n : n \in \mathbb{N})$ be the sequence given by: $$ \ a_n = a_1 + (n - 1)d,\quad a_1,\ d,\ n \in \mathbb N,\quad d\gt a_1,\quad \gcd(a_1,\ d)=1. $$ For all terms of $A$ greater than $\ ...
1
vote
2answers
44 views

Prove by the method of Congruences that 31 \mid $2^{5n}-1

I'm being asked to prove, by method of congruences, that $$ 31\mid2^{5n} - 1$$ Now I'm trying to this via mathematical induction (is this the correct way to go about this?). And the method of ...
5
votes
0answers
60 views

finding very small values of $r + s\sqrt p + t\sqrt q$

For integers $p$, $q$ I'm interested in finding good approximations to $0$ of the form $r + s\sqrt p + t\sqrt q$ for rational $r$, $s$ and $t$. In particular, I'd like to generate approximations ...
0
votes
1answer
30 views

How find the greatest odd number $N$, such for any odd $k(<N)$ if $(N,k)=1$,then $k$ is prime number.

Find the greatest odd numbers $N$,such for any odd $k(<N)$ if $\gcd (N,k)=1$,then $k$ is prime number. It is said the odd $N\le 105$? if $N=5$,then $k=3$ such $\gcd(k,N)=1$,and $k=3$ is ...
3
votes
2answers
55 views

prove $ab\in A$, if $A=\{x^3+y^3+z^3-3xyz\mid x,y,z\in \mathbb Z\}$, $a,b\in A$

let $A=\{x^3+y^3+z^3-3xyz\mid x,y,z\in \mathbb {Z}\}$, prove that: if $a,b\in A$, then $ab\in A$, I think we must find $A,B,C$ such $$A^3+B^3+C^3-3ABC=(a^3+b^3+c^3-3abc)(x^3+y^3+z^3-3xyz)$$ ...
1
vote
1answer
63 views

The relationship between each harmonic numbers

In Knuth's "Concrete Mathematics" in chapter about numbers below equality is given $$H_n = \ln n + \gamma + \frac{1}{2n} - \frac{1}{12n^2} + \frac{\epsilon_n}{120n^4} $$ where $0 < \epsilon_n < ...
2
votes
1answer
78 views

Does the limit of a sequence with floor function exist?

Question : Let $a_n=n\alpha-\lfloor n\alpha\rfloor\ (n=1,2,\cdots)$ where $\alpha$ is an irrational number. Then, does the limit $n\to\infty$ of $(a_n)^n$ exist? I know that ...
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0answers
37 views

What is the maximum possible tile in the popular game “ 2048 ” !? [duplicate]

I know it is possible to know that for the 4x4 game , but , I am NO maths geek , so what is the number !? Also , is it possible to figure that number out using a formula for the general NxN version !? ...
0
votes
2answers
40 views

Question about series.

EDIT: I have deleted the first part of my question, to be honest, I dont know what I was thinking. Thanks for all the answers. My second question is about the equality $$- \sum_{i=1}^{l} ...
0
votes
0answers
50 views

a special function for count

Let be $f:(\mathbb{N} \setminus \left\{0,1 \right\})^2 \rightarrow \mathbb{N}$ function that $f(a,k)=\text{total numbers of }n \in \mathbb{N} \text{ that } \frac{a^n}{n^k} \le 1$ . My question is: ...
0
votes
1answer
12 views

Given $d$, for how many $m$'s is $d$ a quadratic residue mod $m$?

Let $d$ be a fixed, square-free integer, and let $M$ be some very large number. I would like to count the numbers $m \leq M$ such that $m \perp d$ and $d$ is a quadratic residue modulo $m$. Call this ...
2
votes
2answers
73 views

How prove this equation have more one solution $x_{1}+x_{2}+\cdots+x_{n}=d,x_{1}x_{2}\cdots x_{n}=b$

Let $x_{i}\in \mathbb{Z},i=1,2,\ldots,n$, and such that $1\le x_{1}\le x_{2}\le\cdots\le x_{n}$. Show that $$\begin{cases} x_{1}+x_{2}+\cdots+x_{n}=d\\ x_{1}x_{2}\cdots x_{n}=b\\ b,d\in \mathbb{Z} ...
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0answers
26 views

Semi-convergent of continued fractions

I have read this from here The simple continued fraction for $x$ generates all of the best rational approximations for $x$ according to three rules: Truncate the continued fraction, and ...
0
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0answers
26 views

An inequality with $\omega(n)$ [duplicate]

Prove: For any positive integer $k, N$, $$\left(\frac{1}{N}\sum\limits_{n=1}^{N}\left(\omega (n)\right)^k\right)^{\frac{1}{k}}\leq k+\sum\limits_{q\leq N}\frac{1}{q}$$ Where $\sum\limits_{q\leq ...
6
votes
4answers
103 views

Show that a specific $w$ cannot be the root of an quadratic with integer coefficients.

Let $w$ be the only real root of $x^3-x-1=0$. Show that $w$ cannot satisfy the quadratic $ax^2 + bx + c$ ,where $a,b,c\in \Bbb Z$. I have written $$w^3=w+1$$ but I can't go any further than this. ...
2
votes
0answers
51 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) ...
0
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0answers
38 views

Number theory problems relating to Fermats theorem [duplicate]

For all odd primes $p$ show $\left ( \frac{2}{p} \right )= \left ( (-1)^\tfrac{\left ( p^2-1 \right )}{8} \right )$. I have proven that if $q = 2Q + 1$ is prime then $q\mid 2^Q -1$ when $q=8k \pm ...
0
votes
3answers
2k views

Minimum moves to reach destination [closed]

Given that a person is standing at $(0,0)$ and initially look in direction of $X$-axis. Now he can walk only at right angle to previous move. Like if he has to go to $(3,3)$ then $6$ moves are ...
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1answer
39 views

a finite induction question from burton's elementary number theory

this question comes from burton's elementary number theory, 4th edition. question 3 in 1.1 says to use the second principle of finite induction to establish that $$for\ all\ n\ge1,\ ^{(a)}\ ...
0
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1answer
38 views

How to test mathematically if a number contains the highest digit of its radix?

Is there a way to test mathematically if a number contains the highest digit of its radix, and if so how? For example, 101 in base 2 contains the digit 1, highest in base 2; but 101 in base 3 does ...
1
vote
1answer
62 views

Congruences of weights of modular forms modulo primes

I'm trying to prove that for two modular forms $f$ and $g$ of weight $k$ and $k'$ respectively, that are congruent modulo a prime $\ell\ge 5$, their weights are congruent modulo $\ell-1$. This is what ...
0
votes
1answer
129 views

There is no solution for this equation [closed]

Prove that $3^a - 2^a \equiv 0 \pmod a$ for a natural number greater $a>1$ , has no solution .
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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 ...
1
vote
1answer
47 views

If $n$ is a positive integer such that the sum of all positive integers $a$…

I am stuck with the following problem that says: If $n$ is a positive integer such that the sum of all positive integers $a$ satisfying $1 \le a \le n$ and GCD $(a,n)=1$ is equal to $240n,$ then ...
0
votes
1answer
33 views

The sum of the cubes and the amount of combinations.

Quite simply turned out to solve this Diophantine equation, when he made the assumption that the solutions of these equations symmetric. So given this equation: ...
0
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1answer
50 views

Partition numbers with restriction on the greatest part *and* on the number of positive parts

I’m looking at partition numbers. OEIS A008284 says that the number of partitions of $n$ in which the greatest part is $k$, $1 \le k \le n$, is equal to the number of partitions of $n$ into $k$ ...
0
votes
1answer
32 views

Using the euclidean algorithm to find the inverse of $50 \mod 3$

To solve a congruency like $$50x \equiv 17 \mod3$$ You need to find the inverse of $$50x \mod 3$$ For this, you have to write $1$ as a linear combination of $50$ and $3$: $$1 = 50k_1+3k_2$$ ...
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
2answers
67 views

Divisor Pattern - Number Theory

List all positive divisors of $18 $ List all positive divisors of $75 $ Find another number with the same number of divisors. What is the pattern? $18 – 1,2,3,6,9,18 $ $75 – 1,3,5,25,75 $ $99 – ...
3
votes
4answers
64 views

How can I prove that $2^{n+2}\mid(2n+3)!$?

I'm not sure where to proceed or how to go about proving this assertion holds for all natural numbers n: $$2^{n+2} \mid(2n + 3)!$$ The base case is $n=1$, where $2^{1+2}\mid(2\cdot 1+3)!$ which ...
3
votes
3answers
127 views

Number of primes less then $6000$ using $n/ \log n$

So I am trying to use this formula here and is giving me some trouble. If I just substitute $6000$ into the formula, the answer is approximately $1500$. But the number of primes under $6000$ is ...