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.

learn more… | top users | synonyms (1)

11
votes
2answers
328 views

Why is $O_K\otimes \mathbb{Z}_p\cong \oplus_{\mathfrak{p}|p}O_{K,\mathfrak{p}}$?

In my old number theory notebook this is stated as a fact. However, I ran into problems when I tried to prove it. First let me state the (supposed) theorem accurately: Theorem (?) Let $K$ be a ...
5
votes
3answers
314 views

The elliptic curve $y^2 = 23328x^3-890273x^2+14755570x-7^7$

The elliptic curve, $$y^2 = 23328x^3-890273x^2+14755570x-7^7 \tag{1}$$ has the small solution $x = 58$. I know how to find other rational points, but the number of digits in the denominator gets ...
3
votes
4answers
269 views

Factor 90301 without the aid of the computer

Computer break down easily know $90301=73\cdot1237$ Is there any way I want, without the aid of the computer to determine 90301 is a prime number or Composite number
11
votes
2answers
751 views

Thoughts on the Collatz conjecture; integers added to powers of 2

I've had a thought about the Collatz conjecture (the 3n+1 problem). Suppose some number, C, diverges under the iteration. We first note that C must be odd because if C were even it would be halved ...
5
votes
1answer
203 views

A combinatorial number theory proof

How can I prove the following identity: $$\sum_{k=1}^{n}{\sigma_{\ 0} (k^2)} = \sum_{k=1}^{n}{\left\lfloor \frac{n}{k}\right\rfloor \ 2^{\omega(k)}}$$ where $\omega(k)$ is the number of distinct ...
0
votes
1answer
243 views

P-adically Cauchy sequences

I am trying to do the following question Find a p-adically Cauchy sequence which converges p-adically to $-1/6$ in $\mathbb{Z}_7$. In general in $\mathbb{Q}_p$ what is the stronger condition, to be ...
5
votes
1answer
200 views

Exponentiation of a Dirichlet series

I'm trying to understand a proof in Chandrasekharan's Introduction to Analytic Number Theory. Specifically, the proof of the lemma on p.118 before Dirichlet's theorem on primes in arithmetic ...
0
votes
1answer
406 views

P-adic integers

Show that $\frac{2}{p-1}$ is a $p$-adic integer and find its p-adic expansion. P-adic numbers really make little sense to me so any help explaining what to do and why would be really appreciated. ...
1
vote
3answers
624 views

$\mathbb{Q}(\sqrt{d})$ with specific integral basis

I would like some help with the following question. Ireland and Rosen (ch.13#10) For which $d$ does $\mathbb{Q}(\sqrt{d})$ have an integral basis of the form $\alpha, \alpha '$ where $\alpha '$ ...
1
vote
0answers
38 views

Find solutions to a Diophantine Eq. [duplicate]

Possible Duplicate: Methods for quartic diophantine equation Find all solutions of the Diophantine Equation 3x^2 + 2y^4 = z^4 I have just found x=0, y=o, and z=0. I think that I need to ...
9
votes
2answers
581 views

Show that $ a,b,c, \sqrt{a}+ \sqrt{b}+\sqrt{c} \in\mathbb Q \implies \sqrt{a},\sqrt{b},\sqrt{c} \in\mathbb Q $

Assume that $a,b,c, \sqrt{a}+ \sqrt{b}+\sqrt{c} \in\mathbb Q$ are rational,prove $\sqrt{a},\sqrt{b},\sqrt{c} \in\mathbb Q$,are rational. I know that can be proved, would like to know that there is ...
34
votes
5answers
3k views

What is so interesting about the zeroes of the $\zeta$ function

The Riemann $\zeta$ function plays a significant role in number theory and is defined by $$\zeta(s) = \sum_{n=0}^\infty \frac{1}{n^s} \qquad \text{ for } s > 1 \text{ and } s= \sigma + it$$ The ...
5
votes
1answer
216 views

Extending Galois automorphism to group automorphism

Let $F \subset K$ be a field extension of degree $n$, then $F^n \cong K$ as $F$-vectorspaces. Now $K^\times$ acts on $F^n$, by multiplication on $K$, and so $K^\times$ embeds into $GL_n(F)$, and every ...
3
votes
0answers
376 views

A different approach to the strong Goldbach conjecture?

Consider the set A of prime numbers $p_i$ such that $p_i+6$ is not prime (listed in OEIS 140555; see comments thereto). Let 'Goldbach representation' mean a pair of odd prime numbers which sum to a ...
1
vote
2answers
659 views

Number equal to the sum of powers of its digits

I've got another interesting programming/mathematical problem. For a given natural number q from interval $[2; 10000]$ find the number $n$ which is equal to sum of $q$-th powers of its digits, ...
0
votes
1answer
127 views

How to solve this problem?

I want to solve this problem, but I'm stuck. Can anyone can help me out? $$S = \left(\sum_{k=1}^N k^xx^k\right)\bmod M$$ How can I find $S$ if $N$ , $M$ and $x$ are given. Also, $1 ≤ N$, $M ≤ ...
6
votes
1answer
321 views

Prime number $p=4k+1,\;k\in\mathbb{Z}$

Let $p=4k+1,$$p$ is a prime numbe,and $ k\in\mathbb{Z}$. Prove the existence of positive integer $a_1,a_2,\ldots,a_k$ and $b_1,b_2,\ldots,b_k$. $$ ...
2
votes
2answers
254 views

A slightly stronger version of the Goldbach conjecture?

The strong Goldbach conjecture postulates that every even number can be expressed as the sum of two primes. An even number that is just twice a prime is considered a valid exemplar. For low values ...
0
votes
1answer
94 views

euler fermat and primes

Given $n\in\mathbb{N}$ we can write for $n>1$ : $n=p_1^{a_1}\cdots p_s^{a_s}$ with primes $p_i$. Define $k:=lcm(\varphi(p_1^{a_1}),\ldots,\varphi(p_s^{a_s}))$ ( lowest common multiple) I have to ...
2
votes
1answer
191 views

Identity, Bernoulli number

I am trying to prove the following for a very, very long time: $$ 2k=\sum_{j=1}^k \binom{2k+1}{2j}2^{2j}B_{2j} $$ Here, $B_{2j}$ are Bernoulli numbers. I would be extremely happy if somebody could ...
2
votes
1answer
109 views

inequality proof of $x^{y-1} \ge xy$

How to prove $x^{y-1}\geq xy$ with $x,y\in \mathbb{R}$ with $x,y\geq 3$ . Do I need induction? Or is there an elegant way?
1
vote
0answers
132 views

expectation of vector

Let vector $c\in 2N $ is such that first $m$ of its coordinates are $1$ and the rest are $0$ ($c=(1,\ldots, 1,0, \ldots, 0)$). Let $\pi$ be $k$-th permutation of set $\{1, \ldots, 2N\}$. Define ...
3
votes
1answer
245 views

A question about modular forms in SAGE

I was trying to solve Exercise 1.4.5 in Alvaro Lozano-Robledo's book Elliptic Curves, Modular Forms and Their L-functions, which is about representations of integers as sums of 6 squares and its ...
1
vote
1answer
79 views

What is the smallest $n \in \mathbb{N}$ such that $n$ is divisible by $2,3,5$, is square and a fifth power

Can anyone help me prove what is the smallest $n \in \mathbb{N}$ such that $n$ is divisible by $2,3,5$, is square and a fifth power I have so far, for $n,y,q,p,z\in \mathbb{N}$ $n=30q$ , $n=y^2$ ...
4
votes
1answer
98 views

Find $k$-tuples satisfies $j=n_2+2n_3+\cdots+(k-1)n_k$ if $n_1+\cdots+n_k=n$.

Let $n_i \in N$, $i=1,\ldots,k$ and such that $n_1+\cdots+n_k=n$. Fix $j \in N$. I would like to find all $k$-tuples (or algorithm how to find $k$-tuples) satisfies $$ j=n_2+2n_3+\cdots+(k-1)n_k $$ ...
2
votes
0answers
98 views

references for an arithmetic function

I was wondering if anyone is aware of any existing literature on the arithmetic function defined as $$f(n):=2^{\omega(n)}\tau(n).$$ Here $\omega(n)$ is the number of distinct prime divisors of $n$ and ...
6
votes
1answer
303 views

Does this $\zeta(s)$ identity have a name?

I have generalized the product from this thread: Let $s=2n+1$ for $n\ge1$, $$\zeta (s)=\frac{\zeta (2 s)}{\zeta (2)} \prod _{n=1}^{\infty } \frac{\sum _{i=0}^{s-1}(-p_n){}^i}{(p_{n}-1)p_{n}^{s-2}}$$ ...
1
vote
1answer
125 views

The structure of $(\mathbb{Z} / 72 \mathbb{Z})^*$

I am trying to do the following question in preparation for my number theory exam. Write down the structure of $G = (\mathbb{Z}/ 72 \mathbb{Z})^*$ as a product of cyclic groups and find a set of ...
3
votes
2answers
208 views

Solving a Diophantine Equation

Let $p \equiv q \equiv 3 \pmod 4$ for distinct odd primes $p$ and $q$. Show that $x^2 - qy^2 = p$ has no integer solutions $x,y$. My solution is as follows. Firstly we know that as $p \equiv q \pmod ...
0
votes
1answer
83 views

How does one show that $(n-2)! = 2^{m-1} m! (n - 2m)!$ has only one solution for $n\ne 6$?

The obvious solution is 1, for $n=6$ there is another one - $m=3$. How does one show that for other $n$ there are no solutions but $m=1$? This is to show that for $n\ne 6$ all automorphisms of $S_n$ ...
8
votes
3answers
4k views

Calculating the Zeroes of the Riemann-Zeta function

Wikipedia states that The Riemann zeta function $\zeta(s)$ is defined for all complex numbers $s \neq 1$. It has zeros at the negative even integers (i.e. at $s = −2, −4, −6, ...)$. These are ...
2
votes
1answer
148 views

The digit base and the NTT convolution

Suppose I'm using a number theoretic transform (NTT) in an integer field $GF(p)$. I assume that $2n$-th root of unity exists for such a $p$, and I want to compute a convolution of two $n$-length ...
10
votes
1answer
496 views

Does this show that the Apery Constant is transcendental?

Last August I posted this on mathoverflow: http://mathoverflow.net/questions/71856/a-serendipitous-riemann-identity. I show the (slightly revised) equation below: $$\zeta (3)=\frac{2\pi^4}{315} ...
3
votes
1answer
432 views

Pyramid of numbers: reprise

Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed. Suppose we have a triangle of numbers. Atop the triangle is 1: ...
3
votes
2answers
109 views

Looking for a database of results in number theory

Is there a public database in this world consisting of known number fields, their discriminants, and their ideal class groups, etc? If so, how does a lay person like me have access to this database?
6
votes
1answer
182 views

Constructing Magic Squares over $\mathbb{Z}$ from Magic Squares over $\mathbb{Z}_m$

A magic square over $\mathbb{Z}$ is an n x n matrix whose entries are $\{1, \ldots, n^2\}$, with the sum of every row and column identical (in particular, my magic squares are all normal, but the sum ...
2
votes
1answer
147 views

Discrete log of consecutive numbers

I am trying to understand the relation of additive and multiplicative structure of finite field of prime cardinality. Let's say I have a set that behaves nice additively - an interval $I = ...
0
votes
1answer
54 views

In the QS method of integer factoring, how does one know what number can be in the factor base?

I'm not talking about the size of the factor base, but which primes are candidates (only half are). Isn't there some easy way to test a prime to see if it can be?
1
vote
2answers
119 views

Solutions to $x_1+2x_2+3x_3+4x_4+5x_5+6x_6+7x_7+8x_8+9x_9+10x_{10}\equiv0\mod11$

How many solutions does the following equation have: $x_1+2x_2+3x_3+4x_4+5x_5+6x_6+7x_7+8x_8+9x_9+10x_{10}\equiv0\mod11$ where $x_{1...9} \in \{0,1,2,3,4\ ...\ 8,9\}$ and $x_{10}\in\{0,1,2,3,4\ ...
4
votes
2answers
300 views

How often is an irreducible polynomial irreducible?

The question doesn't of course make sense as written in the title. Here is what I really mean: Given a global field $k$ and an irreducible polynomial $P \in k[x]$ Is it true that $P$ is reducible ...
7
votes
2answers
311 views

Continued fractions

I'd really love with concluding that for given integers $a_0,a_1,...a_N$ with $a_i>0$ for $i>0$, representing the continued fraction $[a_0; a_1,....,a_N]$, with the following recursion: ...
1
vote
1answer
436 views

Is there a winning strategy for this game?

Andy and Bob play a game using a long straight row of squares, alternating turns. When it’s Andy’s turn, he writes an A in one of the blank squares. When Bob takes a turn, he writes a B in some blank ...
2
votes
1answer
128 views

$\left|{\sin(\pi \alpha N)}/{\sin(\pi \alpha)}\right| \leq {1}/{2 \| \alpha \|}$

How does one prove the following inequality? $$\left|\frac{\sin(\pi \alpha N)}{\sin(\pi \alpha)}\right| \leq \frac{1}{2 \| \alpha \|}$$ Here $\| \alpha \|$ denotes the distance to the nearest ...
7
votes
1answer
947 views

Divisor summatory function for squares

The Divisor summatory function is a function that is a sum over the divisor function. $$D(x)=\sum_{n\le x} d(n) = 2 \sum_{k=1}^u \lfloor\frac{x}{k}\rfloor - u^2, \;\;\text{with}\; u = \lfloor ...
0
votes
2answers
236 views

Prove Sequences are Uniformly Distributed Modulo 1

Can anybody show me the following Proof? Prove that if $X_n$ for $n=1,2,\ldots,\infty$ is a real sequence that is uniformly distributed modulo $1$, and if $Y_n$, $n=1,2,\ldots,\infty$ is a real ...
2
votes
1answer
296 views

Strong approximation in function fields

How does the strong approximation theorem for global function fields looks like? For the number field $\mathbb{Q}$ it can be expressed as the surjection $$ \mathbb{Q}^\times \times \mathbb{R}^\times ...
2
votes
1answer
135 views

Coset representatives for $\mathcal{O}_K/(\alpha)$.

Let $\mathcal{O}_K$ be the ring of integers of a quadratic extension of $\mathbb{Q}$, and $\alpha$ some nonzero integer. I recently asked why $N(\alpha)=|\mathcal{O}_K/(\alpha)|$, and KcD seemed to ...
6
votes
1answer
158 views

Proof of $\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\frac{\sigma(k)}{k}=\zeta(2)$

Numerically, they seem to be the same, but can we prove that $$\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\frac{\sigma(k)}{k}=\zeta(2),$$ where $\sigma(k)$ is the sum of divisors of $k$.
0
votes
3answers
75 views

If $\log_{b}N$ is rational, what are the limitations on the possible values of $b$ and $N$?

If $\log_{b}N$ is rational, is there a set of values to which $b$ and $N$ must belong? Is there a set of values to which $b$ and $N$ cannot belong? Further, if it is presupposed that $b$ and $N$ are ...
2
votes
2answers
293 views

Are absolute Galois groups compact topological groups

Let $T$ be a finite set of primes and let $K$ be the maximal extension of $\mathbf{Q}$ unramified outside $T$. We have three Galois groups: $G_{\mathbf{Q}} = ...