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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31 views

Two closest sums of pairs of reciprocals

Trying to obtain a better bound for a problem from this bounty question, I obtained the following problem. Let $n\ge 3$ be a natural number. The problem is to estimate (in particular, asymptotically) ...
2
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1answer
323 views

Proof of Andrica when Assuming Oppermann

Proof of Andrica's conjecture by assuming Oppermann's conjecture. Oppermann's conjecture: $$n\geq2\wedge\pi\left(n^{2}-n\right) < \pi\left(n^{2}\right) < \pi\left(n^{2}+n\right).$$ ...
15
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1answer
177 views

The n-th prime is less than $n^2$?

Let $p_n$ be the n-th prime number, e.g. $p_1=2,p_2=3,p_3=5$. How do I show that for all $n>1$, $p_n<n^2$?
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2answers
52 views

Minimum number of moves in Chocolate Puzzle

I've seen this problem on an algorithms competition and although there is an explanation on the website, I couldn't understand it. The abridged problem statement is as follows: Suppose you have two ...
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1answer
31 views

let $n=x^2+y^2$, then all the prime factors of $n$ congruent to $3$ modulo $4$ occur to an even exponent

let $n=x^2+y^2$, then all the prime factors of $n$ congruent to $3$ modulo $4$ occur to an even exponent. I know how to prove the other direction. But this direction seems to be more difficult, and I ...
0
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1answer
41 views

k different numbers, choose n numbers from them where repetition is allowed. [closed]

Suppose I have k different no.s. Each number is available any number of time. Now I want to have n no.s. (so, obviously I can select any no.s from those k for any no. of time). How many ways are there ...
1
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0answers
33 views

Using a Gauss sum to show that $p$ is of the form $x^2 + xy +3y^2$ if and only if $p \equiv 1, 3, 4, 5, 9 \pmod{11}$

Let $p \neq 11$ be an odd prime, and $\zeta$ an $11$th root of unity. Let $g$ be the Gauss sum $$g = \sum\limits_{i=1}^5 \zeta^{i^2} = \zeta + \zeta^4 + \zeta^9 + \zeta^5 + \zeta^3$$ We may ...
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0answers
28 views

Ramification: Riemann surfaces vs Number fields

I am trying to understand the connection between Riemann surfaces and number fields. I am wondering if there an inconsistency in the definition of ramification in terms of Riemann surfaces vs number ...
3
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0answers
30 views

the Teichmüller character

Let $d \geq 2$ be an integer, $K$ a number field containing the $d$-th roots of unity $\mu_d(\mathbb{C})$ and $\mathfrak{p}$ a prime ideal of $K$ not dividing $d$. Let $\mathbb{F}_q$ be the residue ...
5
votes
5answers
154 views

Showing uniqueness of integers in base 3

I have recently begun self-studying Number Theory, and am working on proving: Show that every integer $n>0$ can be uniquely written as $$n = \sum_{i=0}^mc_i3^i$$ where $c_i \in \{ -1,0,1\}$ and ...
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2answers
18 views

Doubt regarding equating coefficients of polynomial in modulo p

I was reading a book and I came across an argument like the following - $$\sum_{i=0}^na_ix^i \equiv \sum_{i=0}^nb_ix^i \mod p$$ $$\implies a_i\equiv b_i \mod p \forall i$$ How to prove this ? I ...
0
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2answers
61 views

How can i find summation of the series $i^k$ [duplicate]

Series : $$\sum_{i =1}^{n} i^k= 1^k+ 2^k + 3^k + 4^k +\ldots+n^k$$ where $k$ is a constant. This does not seem to be Geometric progression , how can I evaluate the sum? If possible if also want to ...
1
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0answers
57 views

About RSA-Factoring

Respected All Today I do not ask any problem on mathematics but only willing to know about the following. According to RSA website found here it is declared that RSA factoring challenge is no longer ...
7
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1answer
143 views

How prove such that $2^n-8$ is divisible by $n$, and $n$ has least three distinct prime factors.

Prove that there are infinitely many postive integers $n$ such that $2^n-8$ is divisible by $n$, and $n$ has least three distinct prime factors. I only find infinitely many postive integers $n$ ...
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0answers
39 views

Conjectured optimal running time for integer factorization

While detecting prime numbers is computationally fast ($O(\log^3 n)$), the fastest known algorithms to split a composite number into its prime factor are very slow (RSA cryptography relies on this ...
1
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1answer
49 views

If 10 is not a solitary number, what properties would a friend of 10 have

It is of course an unsolved problem if 10 is solitary or not, but it is conjectured that it is. (See definition of friendly and solitary number on wiki: http://en.wikipedia.org/wiki/Friendly_number) ...
1
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1answer
36 views

Known upper bound for product of numbers?

Is there a known method for computing the upper bound (in terms of $m$) for the product of some given numbers, $r, s, t, m... \in \mathcal{N} \cup \left\{0\right\}$ when $r + s + t \cdots \leq m$? So ...
4
votes
1answer
160 views

Prime factorizations that yield hyperrectangles with integer diagonals

I have been looking into $n$-dimensional rectangles (aka hyperrectangles) with measures given by any orderless prime-factorization of a natural number, where the diagonal is of an integer length. ...
1
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1answer
38 views

Parts of the whole: Which base begets the largest percentage of fractions expressible as a terminating decimal?

Update: It appears the question I actually meant to ask was quite different. As Robert Israel explained in his answer I was calculating the wrong thing. After writing some ugly code (may take a sec to ...
1
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1answer
49 views

Fast growing hierarchy : How can I show that any sequence grows faster than the one before?

How can I show, that in the fast growing hierarchy, every sequence grows faster than the one before ? A function $f(n)$ is said to grow faster than a function $g(n)$, if for every $k$ there exists ...
2
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3answers
153 views

Divisibility of polynomial

Prove that: $(x^2+x+1) \mid (x^{6n+2}+x^{3n+1}+1) $ and $(x^2+x+1) \mid (x^{6n+4}+x^{3n+2}+1) $. I saw proof in book with third roots of unity but i didn't understand it, so i want to see ...
1
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0answers
33 views

Unramified extension of a imaginary quadratic field.

Let $K$ be an imaginary quadratic number field over $\Bbb{Q}$. Let $K(\sqrt{a})$ be an extension with $a$ an integer. Let $d$ be discriminant of field $K$. Then how to show that $K(\sqrt{a})$ is ...
1
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0answers
56 views

Riemann's explicit formula for $\pi(x)$

Riemann's explicit formula $J(x)=\mathrm{Li}(x)-\sum_{\Im\varrho>0}\left(\mathrm{Li}(x^\varrho)+\mathrm{Li}(x^{1-\varrho})\right)+\int_x^\infty\frac{\mathrm{d}t}{t(t^2-1)\log t}-\log2,$ where ...
5
votes
1answer
40 views

primitive element for a Hilbert class field

I am trying to solve the following problem which I found in a book. Find a primitive element for the Hilbert class field for $\Bbb{Q}(\sqrt{-17})$? Any hints..
3
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1answer
74 views

Showing irrationality of $\zeta(k)$ for some $k$ without calculating the value.

For $s\in (1,\infty)$ let $\zeta(s):=\sum_{n=1}^\infty \dfrac 1{n^s}$. Is there a way to show that $\zeta(2k)$ is irrational for some integer $k\geq 1$ without finding explicit formulae?
1
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1answer
80 views

A question on prime density

Let A = {c > 1 : there exists a natural number m, such that for every n > m, there is a prime between n and cn}. Bertrand's postulate says that A contains 2. My question is : Is inf A = 1 ? If not, ...
2
votes
1answer
27 views

Is the mimal polynomial of an algebraic number separable modulo primes not dividing its discriminant?

Let $K$ be an algebraic number field, $K=\Bbb{Q}(\alpha)$. Let $f \in \mathbb Z[X]$ be the minimal polynomial of $\alpha$. If $p$ does not divide $\operatorname{disc}(f)$, then $f$ mod $p$ is a ...
1
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1answer
37 views

Specific question on imaginary quadratic field [closed]

How to solve the following question?! Let $K$ be an imaginary, quadratic field and let $L/K$ be a Galois extension. If $\tau$ is complex conjugation, show that: (a) $L/\Bbb Q$ is Galois iff ...
5
votes
1answer
103 views

$\left(n^n\right)_b = \left(n\right)_b\left(n\right)_b\ldots\left(n\right)_b$

A friend of mine asked me this today and I was not able to give him an answer. Given a base $b$, find (or show the absence of) an integer $n>1$ s.t. ...
1
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0answers
49 views

Galois group of splitting field of a polynomial

Let $f$ be a polynomial with integer coefficients and irreducible over $\Bbb{Q}$. Let $p$ be a prime. Suppose $f(mod $ $p$) can be written down as a profuct of $r$ distict irreducible polynomials ...
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1answer
72 views

Collatz conjecture can't be disproved by a counter example [closed]

I think that Collatz conjecture can't be disproved by a counter example because you can't know where does the sequence end.
3
votes
2answers
52 views

What is the fastest growing primitive-recursive-function?

Fast growing functions tend to be not primitive-recursive. So I wonder if there is a limit how fast a function can grow, if it is known that it is primitive recursive. What is the fastest growing ...
0
votes
1answer
41 views

What is the slowest growing function that cannot be proven to be total by PA?

I asked the question if PA can prove any function growing faster than $f_{\epsilon_0}(n)$ to be total. The answer was no. What about the converse : Can prove PA every function growing slower than ...
2
votes
2answers
70 views

Prove that $5\mid x\,$ if $\,x, y \gt 1 $ satisfy $2x^2 - 1=y^{15}$

If $x\gt 1$ and $\,y\gt 1,$ with $ x, y \in \mathbb N$ so that $(x,y)$ satisfies the equation $$2x^2-1=y^{15},$$ then prove that $5\mid x$. $\mod {10}$ gave me just what the last digit of y can be.
6
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1answer
93 views

What is the least number $n$, such that $n^{2015}+2015$ is prime?

What is the least number $n$, such that $n^{2015}+2015$ is prime ? According to my calculations, there is no prime for $n\le 6000$. It is clear, that $n$ must be even, since $n^{2015}+2015$ must be ...
3
votes
2answers
65 views

$0$ and $p^n \mathbb{Z}_p$ are the only ideals of $\mathbb{Z}_p$

I am looking at the proof of the sentence: $\mathbb{Z}_p$ contains only the ideals $0$ and $p^n \mathbb{Z}_p$ for $n \in \mathbb{N}_0$. It holds that $\bigcap_{n \in \mathbb{N}_0 p^n \mathbb{Z}_p}=0$ ...
0
votes
1answer
34 views

generator of number field inside a given number ring

If $R$ is a number ring (i.e., a subring of a number field) and $K$ its fraction field. Why can we always find an element $a\in R$ such that $K = \mathbb Q(\alpha)$? Probably I am missing an easy ...
0
votes
1answer
50 views

shuffling cards

I have $N$ cards with numbers written on it (from 1 to N, each card have only one number). Now, I divide them into $2$ halves. Next, I take one card from second half, one from first half and again, ...
9
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3answers
165 views

How prove this $\{a\}\cdot\{b\}\cdot\{c\}=0$ if $\lfloor na\rfloor+\lfloor nb\rfloor=\lfloor nc\rfloor$

Interesting problem Let $a,b,c$ be real numbers such that $$\lfloor na\rfloor+\lfloor nb\rfloor=\lfloor nc\rfloor$$ for all postive integers $n$. Show that: $$\{a\}\cdot\{b\}\cdot\{c\}=0$$ ...
1
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1answer
109 views

Is $123456788910111121314\cdots$ a $p$-adic integer?

On the back of this question comes the natural question of whether the string $$1234567891011121314\!\cdots$$ is even a number at all. While that sort of question is vague, given the lack of generic ...
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0answers
39 views

Prime Space - straight lines only connect primes?

I was reading a very obscure article printed out at my university's library, and there was a topic which I wish to discuss a bit further. That is, the author defined a space, namely $\mathbb{P}$, ...
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1answer
36 views

Irreducibility of multivariate polynomials over algebraic numbers [duplicate]

Why a polynomial in $\bar{\mathbb{Q}}[x_1,\dots,x_n]$ is irreducible iff it's irreducible in $\mathbb{C}[x_1,\dots,x_n]$?
2
votes
1answer
47 views

Binary quadratic forms - Equivalence and repressentation of integers

If $f,g$ are two binary quadratic forms, $f$ and $g$ are equivalent, if there is an integer matrix $M$ with determinant $\pm1 $ such that $G=M^T F M$ where $F,G$ are the matrizes that define $f,g$. It ...
1
vote
1answer
37 views

If $x=[a_0,a_1,\dots]$ show that $\mu$-almost every $x \in (0,1/N]$ is infinitely recurrent

Let $G$ be the Gauss map, $$G(x)= \begin{cases} 0 & \text{if} \ x=0 \\ \{\frac{1}{x} \}=\frac{1}{x} \ \mathrm{mod} \ 1 & \text{if $0<x\leq 1$}\end{cases}$$ and $\mu$ be the ...
0
votes
0answers
32 views

On $p^{\log_q n}$, where $p$ and $q$ are distinct primes

Let $p,q$ be distinct primes, $n>1$ an integer with $\log_q n $ irrational. It was, and probably still is, a conjecture that $p^{\log_q n}$ is non-integer. What progress has been made towards it?
4
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0answers
50 views

Not using Jacobi symbol how to prove For all positive integer $n>1$ $2^n - 1 \not | 3^n-1$?

There is a proof: if $n$ is even,then $3|2^n-1$ but $3\not|\;3^n-1$,It is correct; if $n$ is odd,suppose $2^n-1|3^n-1$,then $3^n \equiv 1(\mod 2^n-1)$,then ...
1
vote
2answers
174 views

Alice and Bob number sum game

Alice and Bob play a game with first $N$ positive numbers. Out of these $N$ integers some $K$ integers are missing. So both decided to play with remaining $N-K$ integers and in this game Alice wants ...
1
vote
1answer
45 views

On the prime number theorem in shorts intervals

In 1988 Heath-Brown (" The number of primes in a short interval ", J. reine angew. Math. 389, 22-63) proved this theorem: Let $\varepsilon\left(x\right)\leq\frac{1}{12}$ be a non-negative function ...
8
votes
2answers
115 views

A result of Erdős on increasing multiplicative functions

Erdős proved that if $f(n)$ is a monotone increasing function from the natural numbers to the positive reals, and $f(n)$ is completely multiplicative, then there exists some constant $C$ such that ...
4
votes
1answer
166 views

Existence of $x$ such that $2^x =a,3^x=b,5^x=c$ for some integers $a,b,c$

Conjecture: There does not exist a non-integer $x$ such that $$2^x=a$$ $$3^x=b$$ $$5^x=c$$ where $a,b,c$ are all integers. I'm aware that the similar question There does not ...