Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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9
votes
3answers
424 views

Is there a (real) number which gives a rational number both when multiplied by $\pi$ and when multiplied by $e$?

Besides $0$ of course. What about addition and exponentiation? I would think there's no such number, but I'm not sure if I could prove it.
1
vote
1answer
442 views

solving the mathematical induction problem

I have been reading the mathematical induction and attempt to solve this given problem Prove that $$2\cdot2^1 + 3\cdot2^2 + 4\cdot2^3 + 5\cdot2^4 +\ldots+ (n+1)2^n = n2^{n+1}\;.$$ Any clues?
-1
votes
1answer
88 views

$\log$ transform of the fundamental theorem of arithmetic? [closed]

Taking the canonical form of the fundamental theorem of arithmetic in the form: $$n=\prod_{j=1}^\infty p^{m_j}_j \qquad ;m_j\in \Bbb N_0$$ Does anybody know about a $\log n$ transform of this? Note: ...
0
votes
2answers
72 views

Why is $\frac{1}{2\pi i} \int_C \left( \frac{x}{n} \right)^s \frac{ds}{s} = \theta(x-n) $?

I'm trying to understand the equation: $$\frac{1}{2\pi i} \int_C \left( \frac{x}{n} \right)^s \frac{ds}{s} = \theta(x-n).$$ Here $x\in \mathbb{R}, x\geq 0$, and $C = \{s:\operatorname{Re}(s) = ...
3
votes
1answer
505 views

Non-integer bases and irrationality

I read somewhere: When it comes to properties like prime, irrational, rational, divisible by 2, etc., nothing changes when you change base. But I'm not sure about the rational/irrational one. ...
1
vote
1answer
46 views

how you show that $[\frac{a}{n} ]^2=1$, where $a \in \mathbb{Z}$ and $n$ is odd integer?

$[\frac{109}{1925} ]=[\frac{109}{5} ]^2[\frac{109}{7} ][\frac{109}{11} ] = [\frac{4}{5} ]^2[\frac{4}{7} ][\frac{-1}{11} ] = (?)^2[\frac{2^2}{7} ][\frac{-1}{11} ] = (?)^2\cdot 1 \cdot ...
2
votes
1answer
95 views

The smallest positive integer $n$ satisfying a given condition

Given any positive integer $g$, what is the smallest positive integer $n$ such that $$\left\lceil \dfrac{(n-3)(n-4)}{12}\right\rceil>g.$$$\lceil x\rceil$ is a ceiling function of $x$.
3
votes
1answer
203 views

Proving equivalences between prime counting functions.

If we have that: $$\theta(x)=\sum_{p\leq x}\log p,$$ and $$\psi(x)=\sum_{n\leq x}\Lambda(n)$$ Where $\Lambda(n)=\log p $ if $n=p^m$ and $\Lambda(n)=0$ in another case. How can I prove that : 1) ...
4
votes
2answers
98 views

Can this sum be simplified to closed form?

$$\sum_{a=1}^{L} L\left\lfloor\frac{(L-a)b}{L}\right\rfloor^2$$ $L,b$ are known, positive integers. Can this be reduced to a closed form? Mathematica isn't helping me out much here.
20
votes
2answers
341 views

How prove that:$\varphi(2)+\varphi(3)+\varphi(4)+\cdots+\varphi(n)\ge\frac{n(n-1)}{4}+1$

Prove that for $n\ge 3$, $$\varphi(2)+\varphi(3)+\varphi(4)+\cdots+\varphi(n)\ge\dfrac{n(n-1)}{4}+1$$ where $\varphi$ is the Euler's totient function I think we must use this ...
3
votes
2answers
305 views

Zeta function and probability

I know that $\zeta(n) = \displaystyle\sum_{k=1}^\infty \frac{1}{k^n}$ (Where $\zeta(n)$ is the Riemann zeta function) But the reciprocal of $\zeta(n)$ for $n$ a positive integer is equal to the ...
10
votes
1answer
307 views

some standard estimates in Yitang Zhang's paper

I'm trying to understand Zhang's paper on prime gaps, but I can't figure out some "standard" estimates for which Zhang omitted details. As a layman in analytic number theory, I really need some hints ...
10
votes
1answer
1k views

What's a BETTER way to see the Gauss's composition law for binary quadratic forms?

There is a group structure of binary quadratic forms of given discriminant $d$: Let $[f]=[(a,b,c)], [f']=[(a',b',c')],$ where $d=b^2-4 a c=b'^2-4 a' c'.$ The composition of two binary quadratic ...
0
votes
2answers
777 views

Missing Exercises in Elementary Number Theory by Underwood Dudley.

I'm a beginner in math and I just started studying Elementary Number Theory by Dudley. So far I'm impressed, but I've noticed that the book does not include all the solutions to the exercises they ...
6
votes
3answers
132 views

Proving $\sum_{k=1}^m{k^n}$ is divisible by $\sum_{k=1}^m{k}$ for $ n=2013$

I got an interesting new question, it's about number theory and algebra precalculus. Here is the question: a positive integer $n$ is called valid if $1^n+2^n+3^n+\dots+m^n$ is divisible by ...
1
vote
0answers
57 views

Ring of the residual classes $(\Bbb Z/p\Bbb Z)^\times$? $p$-adic integer?

In a recent question we raised the theorem: for a given prime $p$ and a given power $m$ the representation of any positive integer $n\in \Bbb N$ in the form: $$ n=(a_u p - b_u) \; p^m$$ is unique ...
1
vote
1answer
92 views

connect p-adic expansion and fundamental theorem of arithmetic?

On the way to explain a $p$-adic expansion, we consider, when dealing with natural numbers, if we take $p$ to be a fixed prime number, then any positive integer expansion in the form can be written as ...
3
votes
1answer
122 views

Odd part of $n-1$ and primes

Using $n=11$ as an example: ...
7
votes
3answers
6k views

Number of distinct prime factors

Is there a formula that can tell us how many distinct prime factors a number has? We have closed form solutions for the number of factors a number has or the sum of those factors but not the number of ...
4
votes
1answer
158 views

Prove that $10\mid A000793(n\ge16)$

Prove that if $n\ge16,$then $10\mid g(n),$where $g(n)$ is the largest LCM of partitions of $n$. For more information,see http://oeis.org/A000793 Here is the list of $g(n)$ for $n>0,$ ...
2
votes
1answer
220 views

Functional equation for Hecke $L$-series

In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves, Theorem II.10.3, we have Let $L(s,\psi)$ be the Hecke $L$-series attached to the Größencharakter $\psi$. Then $L(s,\psi)$ has ...
5
votes
0answers
157 views

Each new member from the second divides the sum of all previous

All integers from $1$ to $13$ are recorded in a sequence such that each number (from the second onwards) divides the sum of all previous numbers. What numbers can be in the third place and why? I ...
2
votes
1answer
42 views

$X$ is an odd number, $Y$ is a natural number more than 36. If $\frac{1}{X}+\frac{2}{Y}=\frac{1}{18}$, find the set $(X,Y)$?

$X$ is an odd number, $Y$ is a natural number more than 36. If $\frac{1}{X}+\frac{2}{Y}=\frac{1}{18}$, find the set $(X,Y)$ ? Re arranging the given equation, we have, ...
3
votes
1answer
107 views

Is partition function increasing function?

I have some exercises which require knowing the number of partitions of particular numbers, so I used some python code which I found on internet to compute the values of the partition function for the ...
1
vote
1answer
113 views

definition of divisor functions

I have a question about the definition of divisor functions when I was reading primes in tuples by Goldston, Pintz, and Yıldırım: Let $\omega(q)$ denote the number of prime factors of a squarefree ...
2
votes
0answers
31 views

Does the sum over the normed differential of the prime power function equal $2\log2\pi$?

Let $p\in \Bbb P$ a prime and a prime power function: $$\xi_p(x) = p^x$$ with $x \in \Bbb R^+_0$ hence: $$\xi'_p = \frac{d}{dx}\xi_p=\xi_p \log p$$ Taking into account E. Muñoz García and R. ...
9
votes
1answer
850 views

Structure of p-adic units

I am trying to understand the structure of the $p$-adic units. I know that we can write $$\mathbb{Z}_p^\times \cong \mu_{p-1} \times 1 + p\mathbb{Z}_p,$$ where $\mu_n$ are the $n$th roots of unity in ...
3
votes
5answers
195 views

How to (quickly) prove that $24p+17$ is not a square number

Computer says $24p+17$ is not square number for $p<10^7$ so I guess it's not. I know that squares of odd numbers are all $8p+1$ but $24p+17$ passes the test And how to solve problems like this in ...
5
votes
1answer
121 views

Is there a mathematics field that studies the displaying of numbers?

I've read somewhere (not sure where) - that there is a dicotomy between the numbers and the symbols used for representing them, for example: We have the idea of twoness which can be represented in ...
0
votes
5answers
106 views

Finding the largest $n \in \mathbb{N}$ for which $n-7$ divides $n^3-7$

I want to find the largest $n \in \mathbb{N}$ for which $n-7$ divides $n^3-7$. In other words, I am looking for the largest $n$ such that $\frac{n^3-7}{n-7}$ is an integer. Can anyone provide me with ...
1
vote
0answers
47 views

A question about the solutions of a diophantine equation

I would like to know if it's possible to find the solution of the following equation: $$x^k+y^h=z^{kh}$$ in which: $$\{x,y,z\}\subset\mathbb{N}$$ given $k$ and $h$ with: $$\{k,h\}\subset\mathbb{N}$$
1
vote
1answer
173 views

What is the difference between multiplicative group of integers modulo n and a Galois Field

What is the difference between multiplicative group of integers modulo n and a Galois Field? Is $\mathbb{Z}^*_p$ with p prime the same as $GF(p)$? Or is it the same as $\mathbb{Z}/n\mathbb{Z}$? ...
0
votes
1answer
52 views

Limits and digit sums

Let $x\in \mathbb{N}, f(x)$ - digit sum of $x$ , $f^{[n]}(x)$ - digit sum of digit sum of digit sum of $x$.... ($n$ times) $radix = 10$ Does this limits for different $n$ exists and if it's true ...
2
votes
2answers
55 views

which parameters always make this rational equation evenly divisible?

Hi guys I have the following equation: $$x = \dfrac{a + b \times c - b}{c}$$ This is what I know about each variable: $$a \ge 64$$ $$b \ge 0$$ $$8 \le c \le a$$ My questions is there a concise way ...
2
votes
1answer
820 views

How do you prove that there are infinitely many primes of the form $5 + 6n$?

There should be infinitely many primes of the form $5+6n$. How do you prove it? The same should be true for $7+6n$.
2
votes
1answer
93 views

Compactness principle via model theory.

A standard method of getting more concrete results from more abstract ones in Ramsey theory is the so called Compactness Principle. It is best illustrated by example. Here is the standard version of ...
4
votes
2answers
87 views

Choosing $a$ s.t. $\frac{a^k - 1}{a-1}$ is not a prime power

Let us suppose that we are presented with a positive integer $k$ and asked to come up with a positive integer $a$ such that $\frac{a^k - 1}{a-1}$ is not a prime power, or just to prove in an ...
4
votes
1answer
193 views

Legendre Symbol - Find Prime $p$ Which Divides A Polynomial

I need to find a general form of a prime number $p$ which divides the polynomial $x^2-6$, i.e. $p$ such that $x^2 - 6\equiv 0\text{ (mod }p)$. By Legendre symbol, I actually need to find a prime p ...
3
votes
1answer
89 views

Prime number question

Can somebody please give me a hint on how to start this question: Let $a$ and $n$ be two positive integers with $a,n ≥ 2$. Assume that $a^n−1$ is a prime number. Prove that $a = 2$ and $n$ is a prime ...
2
votes
2answers
96 views

Has anyone studied this operator?

I've been studying a particular unary operator on the commutative ring $\mathbb{Z}/n\mathbb{Z}$. The operator is: $\downarrow(x) = y\pmod{n}$ iff $n \equiv y \pmod{x}$, where $0< x,y \le n$. The ...
5
votes
2answers
335 views

Find the rational points on $1 + 18 x + 81 x^2 + 44 x^3 = y^2$ with Sage

I'm trying to use Sage on-line,but I meet some trouble with the code of it. I want to find the rational points on an ellipse curve,such as $$1 + 18 x + 81 x^2 + 44 x^3 = y^2,\tag1$$ I know that ...
27
votes
2answers
715 views

A beautiful limit involving primes and composites

I observed the following limit empirically. Let $p_n$ be the $n$-th prime and $c_n$ be the $n$-th composite number then, $$ \lim_{n \to \infty}\frac{1}{n}\sum_{i=1}^{n}\frac{p_n c_n}{p_n c_n + ...
2
votes
1answer
96 views

Elliptic curve, number of elements of finite order

I've been looking at this problem for some time now and just can't seem to get the right idea. The problem is: Consider the elliptic curve $C:y^2=x^3+bx$ defined over the rational numbers with $b$ a ...
9
votes
1answer
233 views

Last digits of a power of 2

Prove that there exists a power of 2 such that the last 1000 digits in its decimal representation are all 1 and 2. One fact that I think can be used in this problem: if $2^{n}=\cdots dn$ where ...
4
votes
2answers
5k views

This is a possible proof of the Riemann Hypothesis [closed]

http://arxiv.org/abs/1305.6845 The above link claims to have solved the Riemann Hypothesis. It's not mine, of course. I just saw this on Tumblr and realized I needed bigger guns. This proof looks like ...
9
votes
1answer
388 views

Improving Zhang's prime gap

I am referring to Zhang's paper. Since the set $\cal{H}$ is a subset of $[3.5\times 10^6, 7\times 10^7]$, shouldn't the prime gap he obtained be less than $ 7\times 10^7 - 3.5\times 10^6$ rather than ...
9
votes
5answers
403 views

Is it true that for every prime number $p >3$, the next one will be less than $p+p/2$?

I know that for every prime $p$, the next prime is less than $2p$. Can we improve this statement? Can it be less than $(3/2)p$? What is the best function of $p$ for which this is true (for every ...
4
votes
2answers
153 views

Where do $p$-adic numbers and $p$-Sylow theory both appear?

Both $p$-adic numbers and $p$-Sylow theory are by design "arithmetic" ways of "localizing," so it stands to reason they might be in cahoots in certain contexts. Are they?
15
votes
1answer
337 views

How to prove there are no more positive integers that are products of 2 and 3 consecutive numbers?

$6$ and $210$ share the property that both are the products of both two and three consecutive numbers. $6$ is $2\times3$ and $1\times2\times3$ and $210$ is $14\times15$ and $5\times6\times7$. It was ...
5
votes
1answer
171 views

How does one search intelligently for solutions of a Diophantine equation?

Before the proof of Fermat's last theorem, much evidence was accumulated in favor of the conjecture, by using computer searches to prove that a solution would need to have very large values. What are ...