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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5
votes
1answer
75 views

Number made from the first digits of $2^n$

Consider the number c made from the first digits of $2^n$. To be more precise, the n-th decimal digit of c is the first digit of $2^n$. The first digits from c are : ...
1
vote
1answer
42 views

Are the conjectural values of $H_{k}:=\lim\inf_{n\to\infty}p_{n+k}-p_{n}$ available somewhere?

The question is in the title. It can be found on the current Polymath 8b project page that one expects to have $H_{1}=2$, $H_{2}=6$, $H_{3}=8$, $H_{4}=12$ and $H_{5}=16$ but I'm interested in larger ...
1
vote
2answers
50 views

Addition of irrational numbers algorithmically

How to perform addition of irrational numbers algorithmically, in the standard $b$-adic expansion I do not know how to start the "standard" digit-by-digit way of doing this cause irrational numbers ...
0
votes
1answer
53 views

If $\frac{a+b\alpha+c\alpha^2}{3} \in R$, then $a\equiv b \equiv c \equiv 0 \pmod3$

If $\frac{a+b\alpha+c\alpha^2}{3} \in R$, then $a\equiv b \equiv c \equiv 0 \pmod3$ where $R=\mathcal{O}\cap\mathbb{Q}[\alpha]$ I know that if $\frac{a+b\alpha}{3}$, then $a\equiv b \equiv 0 ...
1
vote
1answer
80 views

Proving $\lambda(ab) = \mathrm{lcm}\{\lambda(a),\lambda(b)\}$ where $\lambda(.)$ is Carmichael's Function

In proving : if $a$ and $b$ are coprime integers then $\lambda(ab) = \mathrm{lcm}\{\lambda(a),\lambda(b)\}$ where $\lambda(.)$ is Carmichael's Function where does $a$ and $b$ are coprime is ...
3
votes
3answers
233 views

Find all postive integer numbers $x,y$,such $x+y+1$ divides $2xy$ and $x+y-1$ divides $x^2+y^2-1$

Find all postive integer $x$ and $y$ such that $x+y+1$ divides $2xy$ and $x+y-1$ divides $x^2+y^2-1$ My try: since $$(x+y)^2-2xy=x^2+y^2$$ I know this well know reslut: ...
40
votes
1answer
914 views

Are there infinite many $n\in\mathbb N$ such that $\pi(n)=\sum_{p\leq\sqrt n}p$?

Are there infinite many $n\in\mathbb N$ such that $$\pi(n)=\sum_{p\leq\sqrt n}p,\tag{1}$$ where $\pi(n)$ is the Prime-counting_function? For example, ...
4
votes
2answers
537 views

Ways of making polite numbers?

Given that $n$ is a polite number, meaning that it can be expressed as the sum of two or more consecutive positive integers, how many different ways are there to express $n$ as the sum of at least two ...
7
votes
6answers
1k views

Why every prime (>3) is represented as $6k\pm1$

Why is every prime (>3) representable as $6k\pm1$? Afterall, by putting values of k, we don't just get primes but also composites. Then why not $2k+1$ or $3k+2$ or $4k+1$ etc. Is it because of ...
2
votes
2answers
308 views

Does every normal number have irrationality measure $2$?

A normal number is a number whose digit expansion in any base is "uniform" in the sense that all finite digit strings occur with the "statistically expected" frequency. I read a sentence somewhere ...
1
vote
1answer
106 views

Embedding $\mathbb Q^c$ into $\mathbb Q^c_p$

Let $p$ be a prime number and $\mathbb Q_p$ the $p$-adic completion of $\mathbb Q$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$. Is there an embedding $$j: \mathbb Q^c ...
6
votes
1answer
332 views

Was Fermat's last theorem proved based on Peano's postulates?

Is the proof of Fermat's last theorem solely based on the Peano's postulates $+$ first order logic? Or it contains other axiomatic systems as well? What does it mean from foundations of math ...
2
votes
1answer
64 views

Definition of $\mathbb Q^c_p$

Let $p$ be a prime number and let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q $ in $\mathbb C$, i.e. the field of algebraic numbers. Is it possible at all to define the $p$-adic completion ...
5
votes
4answers
267 views

how many $1$s in the first n digits of $\pi$?

how many $1$s are there in the first n digits of $\pi$? Any good approximation of its distribution? How about the place of the $n$th $1$? Are these two questions related?
2
votes
0answers
51 views

About the finiteness of Sha(A/K)

As this question is pretty vague due to my huge lack of knowledge of the subject, it may not be suitable for MathOverflow, and so I prefer to ask it here. If I well understood what I read on ...
5
votes
1answer
593 views

Rational roots of a cubic polynomial

Find all distinct non-zero rational numbers $a$, $b$ and $c$ such that $x (x+a) (x+b) +c$ has 3 distinct non-zero rational roots. What I have so far: Let the polynomial have factorization ...
1
vote
1answer
37 views

How prove this $n|a$ if $S_{m}-S_{k-1}=\dfrac{a}{b}$

let $$\dfrac{a_{1}}{b_{1}},\dfrac{a_{2}}{b_{2}},\cdots,\dfrac{a_{n}}{b_{n}}$$ be Rational number,and such $$\gcd(n,\prod_{i=1}^{n}b_{i})=1$$define ...
1
vote
1answer
131 views

How prove this such $\frac{w(n+k)}{w(n)}>\alpha,\frac{\Omega{(n+k)}}{\Omega{(n)}}<\beta$

For any positive integer $n$, write $$n=p^{a_{1}}_{1}\cdots p^{a_{l}}_{l},$$ where the $p_{i}$ are prime numbers. Define $$w(n)=l,\quad\Omega{(n)}=a_{1}+a_{2}+\cdots+a_{l} \, .$$ For any given ...
2
votes
2answers
93 views

Number of Solutions

How to to calculate the number of solutions for the equation $A+B-\gcd(A,B)=R$ where we are given $R$ in the question ? In this question we have to calculate the number of combinations of $A$ and $B$ ...
2
votes
1answer
63 views

Projective coordinates for elliptic curves

If we consider an elliptic curve projectively, it is a homogeneous form in $3$ variables say $x$, $y$ and $z$. How is this related to the Thue equations (homogeneous forms in $2$ variables)? I'm ...
0
votes
1answer
41 views

if $\beta\in \mathcal{O}\cap\mathbb{Q}[\mathbb{w}]$, then $\beta=a_{0}+a_1w+…+a_{p-1}w^{p-1}$

$w=e^\frac{2i}{p}$ where p is odd prime. $\mathbb{Z}[]$ if $\beta\in \mathcal{O}\cap\mathbb{Q}[\mathbb{w}]$, then $\beta=a_{0},+a_1w+...+a_{p-1}w^{p-1}$ where $a_i's$ are unique integers this a ...
2
votes
1answer
279 views

How to show that $1,\alpha,\alpha^2/2$ is an integral basis of $R=\mathcal{O}\cap \mathbb{Q}[\alpha]$

How to show that $1,\alpha,\alpha^2/2$ is an integral basis of $R=\mathcal{O}\cap \mathbb{Q}[\alpha]$. ($\mathcal{O}$ is ring of algebraic integers) $\alpha$ is a root of $f(x)=x^3+2x^2+4$ which is ...
2
votes
1answer
50 views

How to show $a^2+2B^2=p$ has integer solutions for all primes p with $(\frac{−2}{p})=1$

How to show $a^2+2b^2$=p has integer solutions for all primes p with $(\frac{−2}{p})=1$ (legendre symbol) Partial solution: $(\frac{−2}{p})=1$ $\Rightarrow$ p $\ |$ ...
4
votes
1answer
97 views

Properties of Arithmetic Functions

I was recently working on arithmetic functions and using Perron's formula to obtain asymptotic estimates. One observation I made was that the Dirichlet series often can be written in terms of the ...
5
votes
2answers
169 views

squares which are not the sum of a square and twice a triangular number

I'm trying to determine conditions on integer squares which cannot be written as a square and twice a triangular [all numbers positive], i.e. integers $n \ge 1$ where there are no integers $a,b \ge 1$ ...
8
votes
1answer
220 views

What is the big picture behind AKS algorithm?

Despite a number of question on AKS algorithm here, there does not seems to anything related to the idea behind it (for those who don't know, AKS primality testing is found in PRIMES is in P). I read ...
2
votes
2answers
234 views

AIME number theory problem (unique factorization domains)

I'd greatly appreciate some help with the following problem, from a mock AIME I took. Compute the largest squarefree positive integer $n$ such that $\mathbb{Q}(\sqrt{-n})\cap \overline{\mathbb{Z}}$ ...
-2
votes
1answer
119 views

Compute the class number of $R=\mathcal{O}\cap\mathbb{Q}[\sqrt{51}]$

What is the class number of $\mathcal{O}_K$ where $K=\mathbb{Q}(\sqrt{51})$. Could you please explain.
9
votes
1answer
214 views

What does this music video teach us about 863?

This delightful animation by Stefan Nadelman depicts "the additive evolution of prime numbers", set to Lost Lander's song "Wonderful World": http://www.youtube.com/watch?v=TZkQ65WAa2Q. (If you haven't ...
50
votes
4answers
760 views

Integers $n$ such that $i(i+1)(i+2) \cdots (i+n)$ is real or pure imaginary

A couple of days ago I happened to come across [1], where the curious fact that $i(i-1)(i-2)(i-3)=-10$ appears ($i$ is the imaginary unit). This led me to the following question: Problem 1: Is $3$ ...
93
votes
15answers
8k views

Infiniteness of non-twin primes.

Well, we all know the twin prime conjecture. There are infinitely many primes $p$, such that $p+2$ is also prime. Well, I actually got asked in a discrete mathematics course, to prove that there are ...
0
votes
2answers
67 views

How to show that either $a^2+5b^2=p$ or $c^2+5d^2=2p$ has integer solutions for all prime $p$ with $(\frac{-5}{p})=1$

How to show that either $a^2+5b^2=p$ or $c^2+5d^2=2p$ has integer solutions for all prime $p$ with $(\frac{-5}{p})=1$ would the fact that $\mathbb{Z}[\sqrt{-5}]$=$\mathcal{O}\cap ...
6
votes
2answers
506 views

Riemann Hypothesis and the prime counting function

This article on the prime counting function mentions that the Riemann Hypothesis is equivalent to the statement $$|\pi(x)-\rm {li}(x)|\le \frac {1}{8\pi}\sqrt {x}\log (x)\text { for all }x \geq 2657 ...
3
votes
1answer
130 views

Can the equation $ax^2+by^2=cz^2$ be solved in integers (excluding trivial solutions)?

Suppose $a,b,c\in\mathbb{N}$ and are each squarefree. Is there a general solution for this equation? I found that for this equation to be soluble in integers there are three necessary and sufficient ...
1
vote
2answers
118 views

The number system

Simple question(or maybe not...): Its quite fascinating that with only 10 symbols (for base 10) we can represent any possible natural number. In particular, it seems that no matter how you arrange any ...
0
votes
1answer
70 views

How to show that every prime ideal 0f R contains a unique prime of $\mathbb{Z}$

Let $K$ be a number field with the ring of integers of $R$. How to show that every prime ideal of $R$ contains a unique prime of $\mathbb{Z}$. I have no idea, could you please help.
2
votes
1answer
69 views

Fundamental unit

Let K is a cubic extension of $\Bbb{Q}$ having only one real embedding in $\Bbb{R}$, then can we find fundamental unit in the ring of integers which is real.
3
votes
2answers
220 views

Knuth's arrow up notation again

Consider the following recursion : $$ a(1)=3! $$ $$ a(n+1) = a(n)! \quad\hbox{for all $n\geq 1$} $$ For a given $n$, how can the number $m$ with $$ 10 \uparrow \uparrow m < a(n) < 10 \uparrow ...
2
votes
1answer
41 views

Uniqueness of points in Elliptic Curve addition

When working on a curve E, is the point yielded by P + Q (some P and Q on E) completely unique? What I mean is there are no other points on E sharing the same x or y value. Thanks!
2
votes
1answer
108 views

Legendre Symbol as Trig Product

I was unable to prove these two identities found as exercises in Chapter 5 of Ireland and Rosen. $$\left(\frac{2}{p}\right)=\displaystyle\prod_{j=1}^{(p-1)/2}2\cos\left(\frac{2\pi j}{p}\right)$$ and ...
4
votes
1answer
123 views

Dirichlet series experiment - computing the rational coefficient

Let consider the sequence of numbers $a_n = 0,1,-1,0,1,-1,0,1,-1, ...$ extended periodically ( so it has period $9$, $a_{n+10}=a_n$. In fact, this is a Dirichlet character $a_n = \chi_9(n)$ modulo 9. ...
3
votes
1answer
107 views

Irrational number?

Is the solution of the equation $$x + \arctan(x) = \pi$$ irrational ? The equation of $x + \arctan(x) = 1$ must be transcendental because for any nonzero algebraic $x$, $arctan(x)$ is ...
13
votes
1answer
423 views

Prove that this number is irrational

The number $a=0.12457...$ is defined as follows: The digit on the $n$-th place after the dot is the first digit left to the dot of the number $n\sqrt2$. For example, for $n=1$ we have ...
1
vote
2answers
110 views

Find a permutation of this expression to obtain a maximum possible value

Find a permutation $a_1, a_2, ..., a_{1001}$ of the numbers $1,2,...,1001$ such that the expression $$a_1^{a_2^{a_3^{...^{...^{a_{1001}}}}}}$$ take a maximum possible value.
38
votes
9answers
1k views

Find all positive integers $n$ s.t. $3^n + 5^n$ is divisible by $n^2 - 1$

as is the question in the title, I am wishing to find all positive integers $n$ s.t. $3^n + 5^n$ is divisible by $n^2 - 1$. I have so far shown both expressions are divisible by $8$ for odd $n\geq 3$ ...
2
votes
2answers
56 views

Is there another way to represent this summation?

I wish to calculate $\sum_{x=1}^{n}\sum_{y=1}^{n} f(x,y)$ where $x>2y$. I can do this by changing $y$'s upperbound to the floor of $(x-1)/2$ but this makes simplification of the summation harder ...
4
votes
4answers
189 views

Solve $x^2+x+3=0$ mod $27$

I was preparing for my Number Theory class for next semester and one of the questions that I came upon is to solve $x^2+x+3=0$ mod $27$. I have seen modular arithmetic before but never one that ...
1
vote
1answer
86 views

How to prove continued fraction convergents of a number

Let $x=1+\sqrt{3}$. Prove that in pairs the continued fraction convergents of $x$ are $a_n$/$b_n$ < x < $c_n$/$d_n$ where $a_1$ = 2, $b_1$ = 1, $c_1$ = 3, $d_1$ = 1, $a_{n+1}$ = 2$c_n$ + $a_n$, ...
2
votes
1answer
70 views

How many prime ideals in $\mathbb{Z}[i]$ contain $10$?

This is an exam question I don't think I got. It asked how many prime ideals in $\mathbb{Z}[i]$ contain $10$. I know the prime ideals in $\mathbb{Z}[i]$ are principal ideals generated by primes, and ...
1
vote
2answers
50 views

Proof about GCD's

Prove that if $a, b$ and $c$ are integers with $b \neq 0$ and $a=bx+cy$ for some integers $x$ and $y$, then $\text{gcd}(b,c) \le \text{gcd}(a,b).$ I don't understand how to show (b,c) is less than ...