Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.
28
votes
0answers
697 views
All polynomials with no natural roots and integer coefficients such that $\phi(n)|\phi(P(n))$
Let $P(x)$ be a polynomial with integer coefficients such that the equation $P(x)=0$ has no positive integer solutions. Find all polynomials $P(x)$ such that for all positive integers $n$ we have ...
20
votes
0answers
392 views
Generalizing Ramanujan's proof of Bertrand's Postulate: Can Ramanujan's approach be used to show a prime between $4x$ and $5x$ for $x \ge 3$
Perhaps, I've been thinking too long about Ramanujan's proof, but it appears to me that his argument can be generalized beyond $x$ and $2x$. My argument below attempts to show that for $x \ge 1331$, ...
19
votes
0answers
291 views
Computing (on a computer) the first few (non-trivial) zeros of the zeta function of a number field
If $K$ is a number field, whose Galois closure over the rationals has degree 24 or so, and whose discriminant is around $163^4$, then what is a numerically efficient way of computing the first few ...
14
votes
0answers
268 views
Definitive source about Dirichlet finally proving the Unit Theorem in the Sistine Chapel
There is a remark one can find in various books or survey articles (e.g., page 49 of Helmut Koch's "Number Theory: Algebraic Numbers and Algebraic Functions") saying Dirichlet figured out a proof of ...
13
votes
0answers
177 views
Can every number be written as a small sum of sums of squares?
In a practice for a programming competition, one problem asked us to find the smallest number of pyramids which can be built using exactly $n$ blocks, where pyramids have either $k\times k, ...
12
votes
0answers
205 views
Direct proof that for a prime $p$ if $p\equiv 1 \bmod 4$ then $l(\sqrt{p})$ is odd.
Definition: Assume $p$ is a prime. $l(\sqrt{p})=$ length of period in simple continued fraction expansion of $\sqrt{p}$.
The standard proof of this uses the following:
$p$ is a prime implies $p ...
12
votes
0answers
198 views
A local-global problem concerning roots of polynomials
Let $f(x)$ be a polynomial with integer coefficients, irreducible over the integers. Suppose that for all primes $p$, $f$ has a zero in the field $\mathbb{Q}_p(\sqrt{2})$. Here $\mathbb{Q}_p$ denotes ...
12
votes
0answers
295 views
Dedekind Sum Congruences
For $a,b,c \in \mathbb{N}$, let $a^{\prime} = \gcd(b,c)$, $b^{\prime} = \gcd(a,c)$, $c^{\prime} = \gcd(a,b)$ and $d = a^{\prime} b^{\prime} c^{\prime}$. Define $\mathfrak{S}(a,b,c) = a^{\prime} ...
12
votes
0answers
767 views
Proof of Legendre's theorem on the ternary quadratic form
Theorem (Legendre): Let $a,b,c$ coprime positive integers, then $ax^2 + by^2 = cz^2$ has a nontrivial solution in rationals $x,y,z$ iff ...
11
votes
0answers
158 views
Any proof to $\pi^{e}$'s irrationality?
I've searched for this for a while but get nothing...
There are plenty of proofs to irrationality of $e$,$\pi$,$e^{\pi}$. However, I can't find a proof for $\pi^e$. More, when searching for this I ...
10
votes
0answers
164 views
Two questions about weakly convergent series related to $\sin(n^2)$ and Weyl's inequality
By using partial summation and Weyl's inequality, it is not hard to show that the series $\sum_{n\geq 1}\frac{\sin(n^2)}{n}$ is convergent.
Is is true that ...
10
votes
0answers
245 views
$f(x)=\sum_{t}{x \choose t}{n-x \choose k-t}$ - even or odd?
The following function popped in my research:
$$f(x)=\sum_{\array{0\le t\le k \\ t\equiv_p a}}{x \choose t}{n-x \choose k-t}$$
Where:
n,k are natural numbers and $k\le n$.
t is taken over all ...
10
votes
0answers
185 views
An estimate for relatively prime numbers
Fix a finite collection of distinct prime numbers $(p_1, p_2, \dots, p_s)$, denote their product by $N$. For a natural number $n$ let $\beta(n)$ be the number of $k$, $k\leq n$, for which $k$ and $N$ ...
9
votes
0answers
160 views
Curves and Sums-of-Powers Representations
Jacobi first noticed the connection between the functions that bear his name and counting the representations of sums-of-squares,
\begin{eqnarray}
\theta_{3}^{n}(q) = \left( \sum_{k \in \mathbb{Z}} ...
8
votes
0answers
59 views
Series of Cyclotomic polynomials
How can I show that if $\Phi$ is a Cyclotomic polynomial, $$\Phi_n(x)=\prod_{1\leq k\leq n}_{(n,k)=1}(x-\zeta_n^k)$$
With $\frac{d}{dx}\Phi_n(x)=\Phi'_n(x)$
Then, ...
8
votes
0answers
119 views
Combinatorial Interpretation of a Certain Product of Factorials
Let $\mu$ denote the Moebius function. What is a combinatorial interpretation of the following integer,
\begin{align}
\prod_{d \mid n} d!^{\,\mu(n/d)},
\end{align}
where the product is taken over ...
8
votes
0answers
97 views
Homogeneous forms of degree $n$ in $n$ indeterminates over $\mathbb{Z}$: which ones come from the norm of a number field?
Is there a characterization of the homogeneous forms of degree $n$ in $n$ indeterminates over $\mathbb{Z}$ which occur as the norm of some algebraic number ring with a suitable $\mathbb{Z}$-basis?
...
8
votes
0answers
114 views
On the set of integer solutions of $x^2+y^2-z^2=-1$.
Let
$$
\mathcal R=\{x=(x_1,x_2,x_3)\in\mathbb Z^3:x_1^2+x_2^2-x_3^2=-1\}.
$$
The group $\Gamma= M_3(\mathbb Z)\cap O(2,1)$ acts on $\mathcal R$ by left multiplication.
It's known that there is ...
7
votes
0answers
154 views
Prove that $\Sigma_{i=0}^{k} p^{2i}$ ($p$ is prime) is never a perfect square
Prove that
$$
\Sigma_{i=0}^{k} p^{2i}
$$
where $k > 0$ and $p$ is an arbitrary prime, is never a perfect square. I think you can prove it by letting $q = \Sigma_{i=0}^k a_ip^i$, then expanding ...
7
votes
0answers
67 views
Numbers represented by a cubic form
EDIT, April 11, 2013: See answer at http://mathoverflow.net/questions/127160/numbers-integrally-represented-by-a-ternary-cubic-form/127295#127295
This is part 2 ( of 25 discriminants of class number ...
7
votes
0answers
314 views
Why does this identity equal the number of primes?
Can someone explain why this identity gives the number of primes? I don't understand it.
$D_{0,a}(n) = 1$
$D_{k,a}(n) = \displaystyle\sum_{j=1}^{k} \binom{k}{j}\sum_{m=a+1}^{\lfloor ...
7
votes
0answers
110 views
Numbers of the form $(p_{1}^{\alpha_{_{_1}}})^{2}+(p_{2}^{\alpha_{_{_2}}})^{2}+\cdots+(p_{n}^{\alpha_{_{_n}}})^{2}=(p_{m}^{\alpha_{_{_m}}})^{2}$
I'm looking for numbers of the form
$$(p_{1}^{\alpha_{_{_1}}})^{2}+(p_{2}^{\alpha_{_{_2}}})^{2}+\cdots+(p_{n}^{\alpha_{_{_n}}})^{2}=(p_{m}^{\alpha_{_{_m}}})^{2}$$
where $p_{i}$ are prime numbers, ...
7
votes
0answers
109 views
CFT via Brauer groups vs via ideles
I am interested in the relationship between the following two versions of CFT:
Version 1: (Brauer Group Version)
Let $K$ be a number field. One constructs, for every finite place $v$ of $K$, a map ...
7
votes
0answers
147 views
Equidistribution of roots of prime cyclotomic polynomials to prime moduli
Here is a relevant - and longstanding, I'm told - conjecture. Let $f \in \mathbb{Z}[x]$ be irreducible and of degree > 1. Set
$E_p = \{x/p \: | \: 0 \leq x < p, f(x) \equiv 0 \: (p) \}$ = { ...
7
votes
0answers
121 views
Minimal distance limit problem
Consider a square $\{(x,y): 0\le x,y \le 1\}$ divided into $n^2$
small squares by the lines $x = i/n$ and $y = j/n$. For $1\le i \le n$, let $x_i = i/n$ and
$$d_i = \min_{0\le j\le n} \left| ...
7
votes
0answers
284 views
The radical solution of a solvable 17th degree equation
(The question is at the bottom of the post.) Here's a "natural" solvable 17-th deg eqn with small coefficients:
$$\begin{align*}
x^{17}-6 x^{16}&-24 x^{15}-42 x^{14}-31 x^{13}-23 x^{12}-7 ...
7
votes
0answers
144 views
Weak version of Fortune's conjecture
Let $p\#=2\cdot3\cdot5\cdots p$ denote the primorial and $N(x)$ the smallest prime greater than or equal to $x$. Then Fortune's conjecture is that $N(p\#+2)-p\#$ is prime for all $p$. (Heuristic: to ...
7
votes
0answers
130 views
closure of units of number fields in the finite idele topology
Let $K$ be a number field. Denote by $\mathcal O _K^\times$ its rings of units and by $\mathcal O _{K,+} ^\times$ its ring of totally positive units.
Further let us denote by $\mathbb A ...
6
votes
0answers
52 views
Find a parametric formula to $n=(a^2+1)(b^2+1)$ in three distinct ways
I mentioned that the number $4420$ is expressible in the form $(a^2+1)(b^2+1)$ (where $a,b$ are positive integers) in three distinct ways,here is a list of these numbers:
...
6
votes
0answers
50 views
Families of curves over number fields
Is there known a number field $K$ and a curve $F(x, y) \in K(t)[x, y]$ such that $F(x, y)$ does not have points over the field of rational functions $K(t)$ but for all but finitely many positive ...
6
votes
0answers
88 views
modulo of sums of consective powers
I am thinking of whether there is any pattern about sums of consective powers mod $m$.
Assume $m$,$n$,$k$ are integers.
Denote $$f_k(n)=1^k+2^k+\cdots+n^k,$$
The question is:
how does $f_k(n)$ ...
6
votes
0answers
77 views
A formula to calculate summations over all divisors of a fixed integer
I don't know much about number theory, it seems this summation might involve some facts from number theory.
Could you give me some idea of doing it? Thank you very much. The summation is
$$
...
6
votes
0answers
121 views
Question about an upper bound
Let each of the positive integers $a_1 ,\dots, a_n$ be less than $m$ such that the least common multiple of any two of the positive integers $a_1 ,\dots, a_n$ is greater than the integer $m$.
Then ...
6
votes
0answers
154 views
Asymptotic FLT => FLT using ABC Conjecture
Edit:
I'm beginning to suspect I either misread the sources or perhaps something wasn't stated, but it's my guess now that there is no nontrivial way of showing the ABC conjecture implies Fermat's ...
6
votes
0answers
127 views
On Grunwald-Wang theorem
Consider (roughtly speaking) the following statements (the Grunwald-Wang theorem)
Theorem 1 (see here for details Wiki): Let $K$ be a number field and $x \in K$. Then under some conditions : $x$ is ...
6
votes
0answers
183 views
Diophantine special problem
This is my another question on Diophantine equations.
Prove the following great and special problem.
Let $D$ and $k$ be positive integers and $p$ be a prime number such that $gcd(D, kp) = 1$. Prove ...
6
votes
0answers
368 views
How to use Hardy and Wright's text and what corresponding exercises/problem books can I do?
I have just started out with Hardy and Wright's An Introduction to the Theory of Numbers today. I find the lack of exercises in the book as a departure from the style of the textbooks we are so ...
6
votes
0answers
225 views
Question about a proof in Iwaniec-Kowalski's Analytic Number Theory book
My question is about the end of the proof of theorem 1.1, in page 27.
Namely, it is stated that whenever we have a multiplicative function $f:\mathbb{N} \to \mathbb{C},$ let the sequence ...
6
votes
0answers
215 views
Certain permutations of the set of all Pythagorean triples
The fact that the set of all primitive Pythagorean triples naturally has the structure of a ternary rooted tree may have first been published in 1970:
http://www.jstor.org/stable/3613860
I learned ...
6
votes
0answers
100 views
Generalizing Quadratic Reciprocity Law with Dilates
Eisenstein's proof of the Quadratic Reciprocity (QR) (and its Jacobi symbol generalization) both rely on counting lattice points in two congruent triangles. If we take $t$-dilates of these triangles, ...
6
votes
0answers
388 views
What can Euler's identity teach us about (generalised) continued fractions?
We know that $$e^{i \pi} = -1 .$$ We can transform all of the components of this identity into (generalized) continued fractions. When we start of with $\pi$, we see that $$ \Big(3+ ...
5
votes
0answers
82 views
A contest question
$p$ is an odd prime,denote $$f(x)=\sum_{k=0}^{p-1}\binom{2k}{k}^2x^k.$$
Prove that for every $x\in Z$,$$(-1)^\frac{p-1}2f(x)\equiv f(\frac{1}{16}-x)\pmod{p^2}.$$
This is a contest question,I do not ...
5
votes
0answers
91 views
Proof that $t^8+2t^6+4t^4+t^2+1$ is reducible in $\mathbb{F}_p$
Prove that the polynomial $t^8+2t^6+4t^4+t^2+1$ is reducible in $\Bbb F_p$, for all $p\in \Bbb P$.
Here are some examples:
$t^8+2t^6+4t^4+t^2+1=(1 + t + t^4)^2\pmod{2}$
$t^8+2t^6+4t^4+t^2+1=(1 + t) (2 ...
5
votes
0answers
82 views
Hilbert symbol over a ring
Normally the Hilbert symbol over a field $\mathbb{F}$ is defined for $a,b\in\mathbb{F}^*$ as follows:
$$ (a,b)=\begin{cases}1,&\text{ if }z^2=ax^2+by^2\text{ has a non-zero solution }(x,y,z)\in ...
5
votes
0answers
85 views
What's the most efficient algorithm for Divisibility?
What is the most efficient (in time complexity) algorithm known nowadays for the Divisibity Decision Problem: given two integers, say $a$ and $b$, does $a$ divide $b$? Let it be clear that what I ask ...
5
votes
0answers
87 views
What numbers are integrally represented by $4 x^2 + 2 x y + 7 y^2 - z^3$
EDIT: a selection of material relevant to this problem is available at INHOMOGENEOUS
In December 2010 my question appeared in the M.A.A. Monthly, show that $4 x^2 + 2 x y + 7 y^2 - z^3 \neq \pm 2 ...
5
votes
0answers
111 views
Carmichael number factoring
The task I'm faced with is to implement a poly-time algorithm that finds a nontrivial factor of a Carmichael number. Many resources on the web state that this is easy, however without further ...
5
votes
0answers
175 views
Proof of Hardy-Ramanujan inequality in number theory.
On page 3 of http://www.math.dartmouth.edu/~carlp/Lehmer0.5.pdf the author write that the following inequalities follow from "the Hardy-Ramanujan inequality", but he doesn't point to a proof. The ...
5
votes
0answers
116 views
Has the $\Gamma$-like function $f_p(n) = 1^{\ln(1)^p} \cdot 2^{\ln(2)^p} \cdot \ldots \cdot n^{\ln(n)^p} $ been discussed anywhere?
In an older fiddling with the gamma-function (expanding on the idea of sums of consecutive like-powers of logarithms, similarly as the bernoulli-polynomials for the sums of like powers of consecutive ...
5
votes
0answers
112 views
Is there some exact form of $n$ for the number of $(i,j,k)$ satisfying $ijk = (n-i)(n-j)(n-k)$
For any positive integer $n$,
$i,j,k$ are also positive integers, and $0 <i,j,k < n$.
How many solutions of the form $(i,j,k)$ are there for the equation $ijk = (n-i)(n-j)(n-k)$?
