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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2
votes
1answer
77 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 ...
-1
votes
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} ...
0
votes
0answers
25 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
votes
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
50 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
votes
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 ...
0
votes
1answer
38 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
votes
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
60 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
128 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 .
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 ...
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
votes
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
125 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 ...
2
votes
1answer
73 views

Infinite families of prime numbers

What interesting/useful infinite families of prime numbers are there? Right now it would be useful if I could find one with arbitrarily many 1's in its binary representation, but I am doing a larger ...
0
votes
1answer
35 views

Finding out remainder

I need to find the remainder when $ 20!+20^{23} $ is divided by 23 . .please help. Wilson theorem involved , Fermat's little theorem involved as well.
11
votes
0answers
238 views

Which harmonic numbers have prime numerators?

Let $$1+\frac12+\frac13+\cdots+\frac1n=\frac{q}{p}$$ with $(p,q)=1$ and $q$ is a prime number. (I) Prove or disprove that the quantity of $n$ is limited (II) Find all of the $n$ I use the matlab ...
2
votes
1answer
52 views

How to get the numbers that multiplied with $N$ become integers?

How can I get all the numbers (between $0$ and $1$) that multiplied with integer $N$ become integers. Thanks!
4
votes
1answer
235 views

How to prove $\left\lfloor\frac{\sigma{(n-1)}+1}{2}\right\rfloor\le f(n)<\left\lfloor\frac{\sigma{(n-1)}+1}{2}\right\rfloor+n$

Question: let $x_{1},x_{2},\cdots,x_{n}$ be such that $$n\ge 3,x_{1}\le x_{2}\le\cdots \le x_{n},$$ $$x_{1}x_{2}x_{3}\cdots x_{n}=x_{1}+x_{2}+\cdots +x_{n}.$$ Let the number of ordered ...
4
votes
3answers
136 views

Does there exist a $(m,n)\in\mathbb N$ such that $m^3-2^n=3$?

Question : Does there exist a $(m,n)\in\mathbb N$ such that $m^3-2^n=3$? I know that there is no $(m,n)\in\mathbb N$ such that $m^3-2^n=1$ and that there is no $(m,n)\in\mathbb N$ such that ...
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 ...
1
vote
2answers
69 views

The defective doyle

There are six cans each containing doyles(each doyle weighs a gram), out them any of them can be defective. How can we find the defective cans in one weighing? and if each can has only two dozen ...
1
vote
2answers
42 views

3) Consider the finite field Z/31Z. [closed]

3) Consider the finite field Z/31Z. Check that 3 is a primitive root by working out all powers. Deduce from this, quickly, which elements are sixth roots of 1 (not necessarily primitive). Add them up ...
0
votes
0answers
51 views

Probability with dice sum K

Alice rolls a N faced die M times. she adds all the numbers she gets on all throws. What is the probability that she has a sum of K. A N faced die has all numbers from 1 to N written on it and each ...
0
votes
1answer
40 views

Functions mapped to themselves

How many $1$-to-$1$ and onto functions are there for a function $f: A \longrightarrow A$ (i.e. a function mapped to itself)?
1
vote
1answer
64 views

Count pairs with odd XOR

Given an array A1,A2...AN. We have to tell how many pairs (i, j) exist such that 1 ≤ i < j ≤ N and Ai XOR Aj is odd. Example : If N=3 and array is [1 2 3] then here answer is 2 as 1 XOR 2 is 3 ...
1
vote
2answers
38 views

Counting the number of zeros [duplicate]

I am stuck at the question: How many zeros are there when numbers between 1 and 100 are multiplied including 1 and 100, devise some technique for this . Regards
1
vote
1answer
30 views

Can the formula for the sum of the first n squares be deduced from the known value of zeta of 2?

I know this is asking that we swat a fly with a sledge hammer, but this off-the-wall question is prompted by the fact that they both have 6 in the denominator. That is, is that common 6 in the ...
1
vote
1answer
34 views

Transformation property for classical Siegel modular forms of weight 2

Let $\mathbb{H}_g = \{ \tau \in GL_g(\mathbb{C}) | \; {^t\tau} = \tau, Im(\tau) >0\}$ be the Siegel upper half space. There are Eisenstein series $$ E_{2k}(\tau) := \sum_{\gamma\in (P_0\cap ...
-1
votes
1answer
26 views

A question on number theory

For which positive integers $n$ does there exist a positive integer $m_n<n$ such that g.c.d.$(n.m_n)=$g.c.d.$(n,k)$ , $0<k<n \implies m_n=k$ . For example $(4,1)=(4,3)=1,(4,2)=2$ so $n=4$ ...
1
vote
0answers
40 views

Are there infinitely many integers for which $x$ mod $(2n+1)$ is not equal to $n$ for all $n\leq x/3$

I proved the following statement: (all variables are non negative Integers) For all $N$ there exist an $x$ with $x\geq N$ so that for all $n$ from $1$ to floor$(x/3)$ following is true: $x \text{ mod ...
3
votes
1answer
66 views

Determine Units of a Ring $\mathbb{Z}[\alpha]$

I am trying to determine the units of $Z[\alpha]$ where $\alpha$ satisfies the monic polynomial $\alpha^4+\alpha^3+\alpha^2+\alpha+1$. I found $Z[\alpha] := \lbrace a+b\alpha+c\alpha^2+d\alpha^3\;|\; ...
0
votes
2answers
38 views

Show if (m,n) = 1, then for any # p, we have (p,mn) = (p,m)(p,n).

Show that if $(m,n) = 1$, then for any number p, we have $(p,mn) = (p,m)(p,n).$
7
votes
5answers
665 views

Induction hypothesis misunderstanding and the fundamental theorem of arithmetic.

The fundamental theorem of arithmetic is made of two parts: The existence part: There exist primes such that for any natural number $j$, we can write $j$ as a product of prime numbers. The ...
0
votes
0answers
57 views

2 player team knowing maximum moves

Given a list of N players who are to play a game. Each of them are either well versed in a move or they are not. Find out the maximum number of moves a 2-player team can know. And also find out how ...
5
votes
2answers
127 views

Infinitely many prime $n$ for which $n^2 = p + 8$ for some prime $p$.

How to prove that there exist an infinite number of prime $n$ for which $n^2=p+8$ for some prime $p$? Verification of the form $n^2=p+8$ where $n$ and $p$ are some $p$. $$\begin{array}{|c|c|} \hline ...
1
vote
1answer
47 views

Signed determinant of quadratic forms over Q_p

Let $W(k)$ be the Witt-Ring of the field $k$. in this script http://math.uga.edu/~pete/quadraticforms2.pdf at the bottom of page 2 the signed determinant is introduced by $d^\pm (q) = ...
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
1answer
68 views

$\text{lcm}(1,2,3,\ldots,n)\geq 2^n$ for $n\geq 7$

I can prove that $\text{lcm}(1,2,3,\ldots,n)\geq 2^{n-1}$. Newly, i read in a paper that for $n\geq 7$ we have: $$\text{lcm}(1,2,3,\ldots,n)\geq 2^n$$ Can you prove it? (this inequality is an ...