Questions tagged [nt.number-theory]

Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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1 answer
330 views

How can I catalog these generalized Collatz problems?

The Collatz conjecture can be expressed in terms of a ruleset in the language $\{x,+,1,\rightarrow,;\}$: $x + x + 1 \rightarrow x+x+x+1+1;$ $x + x \rightarrow x;$ Whenever a number matches the LHS ...
1 vote
0 answers
69 views

Imaginary quadratic fields with prime class number

Let $K$ be an imaginary quadratic field, with class number equal to an odd prime, say $h_K = p$. In the proof Proposition 2.4 of this paper, Fukuda and Komatsu write, "Since $h_K = p$, there ...
4 votes
0 answers
58 views

Anisotropic semisimple groups with no real compact factor

Let $F$ be a number field, and let $G$ be a semi-simple connected, anisotropic algebraic group over $F$ which is $F$-simple (or almost simple, the question is agnostic to isogenies). Suppose further ...
18 votes
0 answers
348 views

Colouring Gaussian integers according to a numeral system based on powers of $-1+i$

It is easy to check that every Gaussian integer can be written uniquely as a finite sum of the form $\sum_{n\geq 0}\epsilon_n(-1+i)^n$ for ‘digits’ $\epsilon_n$ in $\lbrace 0,1\rbrace$. The sequence $\...
-1 votes
0 answers
91 views

Fermat's theorem on the sums of two squares [closed]

Given the Pythagorean quadruples $d=36*m^2+18*m+4*n^2+2*n+3$ $a=24*m*n+6*m+6*n+1$ $b=2*(3*m+n+1)*(6*m-2*n+1)$ $c=2*(3*m+n+1)$ $a^2+c^2=d^2-b^2=p$ for $n=0$ as $m$ varies we will have potential prime ...
7 votes
1 answer
1k views

Can every finite graph be represented by an arithmetic sequence of natural numbers?

(This is a follow-up to my previous questions Natural models of graphs?.) Erdös in The Representation of a Graph by Set Intersections (1966) states: Theorem. Let $G$ be an arbitrary graph. Then there ...
2 votes
1 answer
319 views

The Fourier transform of the Liouville function?

The Liouville function in number theory is defined as: $$\lambda(n) := (-1)^{\Omega(n)} \text{ where } \Omega(n) := \sum_{p|n} v_p(n)$$ Taking the discrete time Fourier transform and then taking the ...
2 votes
0 answers
149 views

Proof of Remark 6.14(b) of Milne's Arithmetic duality theorems'

Let $E/\mathbb{Q}$ be an elliptic curve.Let $\operatorname{Sha}(E/\Bbb{Q})$ be a Tate-Shafarevich group. Milne's 'Arithmetic Duality Theorems' Remark 6.14(b) describe the following exact sequence. ...
5 votes
2 answers
562 views

Prove positivity of a binomial sum

Some problems appear easy on the face of it, but perhaps they are not. Here is an instance of a certain calculation which is slightly reformulated from its original encounter in a current work. I have ...
10 votes
3 answers
2k views

Non-vanishing of zeta(s), Re(s)=1, without complex analysis?

Say you are allowed to use Fourier analysis, complex variables, Euler-Maclaurin, etc., but no complex analysis - no holomorphic continuations, no definition of analytic function, and, in particular, ...
1 vote
0 answers
55 views

On a numbers $k$ with specific $2$-adic valuation

Let $a(n)$ be A002326 (i.e., multiplicative order of $2 \operatorname{mod} 2n+1$). Let $b(n)$ be A179382 (i.e., the smallest period of pseudo-arithmetic progression with initial term $1$ and ...
5 votes
2 answers
371 views

Difficult elliptic curves for $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$?

Similar to the case $x^4+y^4+z^4 = 1$ discussed in this MO post, define the system, $$x^4+y^4+z^4+1 = (x+y+z+1)^4\tag1$$ $$\frac{x^2+x+1}{(x+y+1)(x+z+1)}=u\tag2$$ $$\frac{y^2+y+1}{(y+z+1)(y+x+1)}=v\...
2 votes
0 answers
97 views

What are the terms given by $\int_0^1\int_0^1 x^py^q \sin(\pi xy) (xy)^{xy} (1-xy)^{1-xy} \, dx \, dy$?

For a positive integer $n$ the terms given by \begin{align} & -\int_0^1 x^n \sin(\pi x) x^x (1-x)^{1-x} \, dx \\[8pt] = {} & \int_0^1\int_0^1 (xy)^n \sin(\pi xy) (xy)^{xy} \frac{(1-xy)^{1-xy}}...
5 votes
0 answers
178 views

Subgroups of ${\rm GL}_{2}(\mathbb{Z}/2^{n} \mathbb{Z})$

Assume that $n \geq 3$. Is there a subgroup $H \leq {\rm GL}_{2}(\mathbb{Z}/2^{n} \mathbb{Z})$ whose order is a power of $2$ so that $\bullet$ $\det : H \to (\mathbb{Z}/2^{n} \mathbb{Z})^{\times}$ is ...
3 votes
1 answer
168 views

Generators of the ideal class group

Theorem 4 of Eric Bach's "Explicit bounds for primality testing and related problems" states the following: Let $K$ be a number field of degree greater than 1. Let $d$ be the absolute value ...
2 votes
1 answer
143 views

Modified Gauss Sum when the characters have different period

Let $\chi$ be a Dirichlet character mod q, and \begin{eqnarray} t(\chi)=\sum_{n=1}^{q}\chi(n)e(\frac{n}{2q}). \end{eqnarray} Do we have a bound or formula for $t(\chi)$ similar to that of the usual ...
3 votes
0 answers
233 views

On thickness of binary polynomials

OEIS A169945 introduces the concept of thickness of a polynomial as the magnitude of the largest coefficient in the expansion of the square of the polynomial. Considering the $2^{n+1}$ polynomials $p(...
1 vote
0 answers
67 views

Kernel of a map of Tate algebras

Let $A$ and $B$ be a pair of Tate algebras over $\mathbb{Q}_p$, and assume $\text{dim}_{\text{Krull}}(B) > \text{dim}_{\text{Krull}}(A)$. Is it true that any morphism $B \longrightarrow A$ must ...
7 votes
0 answers
998 views

Is the Collatz conjecture known to be true for interesting unbounded classes of numbers?

The Collatz or the $3n+1$ conjecture is open. Is there a specific polynomial $f(x)\in\mathbb Z[x]$ whose range is unbounded for which every integer of form $|f(m)|$ at $m\in\mathbb Z$ satisfies $3n+1$...
4 votes
1 answer
827 views

Collatz conjecture and stationarity of time series

The Collatz conjecture is known to all. Has this question been approached by methods related to statistics? I think of Collatz iterates as a time series, and the question of whether we always get the ...
3 votes
1 answer
246 views

Number of Salem–Spencer subsets of $\{1,2,3,\dots ,n\}$

I was wondering about sets that do not contain any $3$-term AP, and came to know that the official name of such a set is Salem–Spencer set. I was considering the question of counting the number of ...
2 votes
1 answer
380 views

Explicit bound on $\sum_{N\mathfrak p \leq x}\chi(\mathfrak p)\ln(N\mathfrak p)$

I'm looking for an explicit bound for $f(x) = \sum_{N\mathfrak p \leq x}\chi(\mathfrak p)\ln(N\mathfrak p)$, where $\chi$ is a Hecke character for a number field $K$ of degree $n$, on the ideals $I_\...
0 votes
0 answers
87 views

Formula for individual term of the Proth numbers

Let $a(n)$ be A080075 i.e. Proth numbers: of the form $k2^m + 1$ for $k$ odd, $m \geqslant 1$ and $2^m > k$. The sequence begins with $$ 3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129 $$...
4 votes
0 answers
150 views

Bezout-type theorem for $p$-adic analytic plane curves

Let $p$ be a prime, and let $f,g \in \mathbb{Z}_p[[x,y]]$ be power series convergent on all of $\mathbb{Z}_p$. Suppose that the intersection of the analytic plane curves cut out by $f$ and $g$ is ...
12 votes
1 answer
505 views

On the equation $9x^3+y^3=z^2+3$

The question is whether there exist integers $x,y,z$ such that $$ 9x^3+y^3=z^2+3. $$ This is one of the nicest (if not the nicest one!) cubic equations for which I do not know whether integer ...
32 votes
3 answers
1k views

Are these fast convergent series for $\log(2)$, $\log(3)$ and $\log(5)$ already known and proven?

Updated on Feb.16.2024 Fortunately for three of these series, Eqs. (1), (3) and (4), I have found a proof using classical methods which I placed in the Answers section below. (I doubt that there is a ...
15 votes
4 answers
987 views

Can we get good rational approximations in all residue classes?

The classic Hurwitz theorem for rational approximations (in simplest form; the constant can of course be improved) gives infinitely many approximations $\frac mn$ to an irrational $\alpha$ with $|\...
0 votes
1 answer
101 views

“Smallest” non-zero linear combination of vectors to obtain a non-negative vector

We say that a vector $\mathbf{x}$ in $\mathbb{Z}^j$ is non-negative if it is of the form \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_j \\ \end{bmatrix} where $x_{j} \geq 0$. Suppose that ...
0 votes
1 answer
114 views

Integer quadratic representation subject to discriminant minimization algorithm

Let $f(x)=ax^2+bx+c$ and $f(x)=n$. Is there an algorithm to choose $a,b,c$ such that the discriminant is minimized? Where $a,b,c,n,x$ are all integers. More concretely, is there an algorithm to find $...
0 votes
1 answer
130 views

Coefficients of 0,1-polynomials factorization

Let $n\in\mathbb{N}$ and $P_{n}$ is $0,1$-polynomial whose coefficients are binary digits of $n$. Let $Q_{1}(x) \cdot Q_{2}(x) \cdots Q_{m}(x)$ - polynomial factorization (over integers) of $P_{n}$. ...
1 vote
1 answer
184 views

How to compute the asymptotic constant for the count of $S_3$-sextic number fields?

I am currently reading this paper counting $S_3$-sextic fields Manjul Bhargava and Melanie Matchett Wood, The density of discriminants of $S_3$-sextic number fields, Proc. Amer. Math. Soc. 136 (2008),...
10 votes
1 answer
458 views

Is 100 the only Leyland number that is a square?

Leyland numbers (named for Paul Leyland) are positive integers of the form $x^y + y^x$ , where x and y are naturals > 1, and also the number 3. The OEIS link is https://oeis.org/A076980 I thought ...
4 votes
4 answers
408 views

A cubic equation, and integers of the form $a^2+192b^2$

This question resembles my previous question A cubic equation, and integers of the form $a^2+32b^2$ , but seems to be more difficult. We are trying to determine whether there are any integers $x,y,z$ ...
0 votes
1 answer
231 views

Product of subspace and its inverse

$\DeclareMathOperator\GF{GF}$Let $R=\GF(q)$ be a finite field with $q=p^r$ elements, where $p$ is a prime number, $S=\GF(q^n)$ be an extension of $R$, where $n\in \mathbb{N}$, $n\geq 2$ and let $K=\GF(...
2 votes
0 answers
191 views

Squares whose differences are squares

EDIT. I've just noticed a thread from 2011 in the "Related" column on the right (click me), where a closely related question is being discussed (the main difference seems to be that, in ...
2 votes
0 answers
174 views

Sets of integers "a little less dense" than the set of prime numbers

Given a set $A \subseteq \mathbb{N}$ of positive integers, put $$ S_A: \mathbb{N} \rightarrow \mathbb{R}, \ \ \ \ N \mapsto \sum_{n \in A \cap \{1, \dots, N\}} \frac{1}{n}. $$ There are obvious ...
8 votes
4 answers
822 views

A cubic equation, and integers of the form $a^2+32b^2$

I am trying to determine whether there are any integers $x,y,z$ such that $$ 1+2 x+x^2 y+4 y^2+2 z^2 = 0. \quad\quad\quad (1) $$ It is clear that $x$ is odd. We can consider this equation as quadratic ...
2 votes
0 answers
64 views

Possible subsequence of the A110978

Let $a(n)$ be A110978 i.e. odd integers that are nonprime, such that there exist two factors of each number that when multiplied together in binary base, do not ever require the use of a "carry&...
1 vote
1 answer
132 views

Density of a set of numbers whose prime factors are defined by congruences

Let $S$ be the set of positive integers not divisible by $3$ where if $p$ is a prime factor of $n \in S$ and $p \equiv 1\bmod 3$ then $p^2$ does not divide $n$, but if $p\equiv2 \bmod 3$ then $p^2$ ...
2 votes
1 answer
354 views

Limit of an infinite series with quadratic arguments

I have encountered a limiting process on some infinite series. So, I would like to ask: QUESTION. Assume $n$ is an even positive integer. Is this true? $$\lim_{r\rightarrow1^{-}}\sum_{j=1}^{\infty}\...
4 votes
0 answers
148 views

Empirical bounds on $\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right|$

It is reasonable to expect that $$\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right| < 2 \log \log t$$ for all $t\geq 4$ (say): a somewhat stronger bound is known for $t\geq 10^{165}$ or so (Theorem 5 ...
0 votes
0 answers
42 views

How to find A(i, d)?

Let $s(n)$ denote the digit sum of a natural number $n$. For $i, d\in \mathbb{N}$ define $$A(i, d) = \limsup_{m\to \infty}\frac{|\{n\leq m | s(n)\equiv i\mod d\}|}{m}.$$ Compute $A(i, d)$ for all $i, ...
2 votes
0 answers
121 views

Prime splitting in the division field of an elliptic curve

Let $E/\mathbb{Q}$ be an elliptic curve with good reduction at two distinct primes $p, \ell$. Suppose the mod $\ell$ Galois representation associated to $E$ is surjective. Let $K=\mathbb{Q}(E[\ell])$ ...
25 votes
1 answer
791 views

Is every polynomial of the form $2x^{2n} -x^n +1$ irreducible over $\mathbb{Z}$?

Is every polynomial of the form $2x^{2n} - x^n +1$ irreducible for $n>0$? Motivation: A few years ago a student asked if $29$ was the largest number which is prime and one more than a perfect ...
1 vote
1 answer
116 views

Iwahori action on the $p$-ordinary line of a principal series representation

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\diag{diag}\DeclareMathOperator\Ind{Ind}\newcommand\Iw{\mathrm{Iw}}\DeclareMathOperator\ord{ord}$Let $F$ be a $p$-adic local field, i.e. a finite ...
0 votes
0 answers
57 views

Upper bound for an additional Product formula

We have three sequences of positive integers $l$, $p$ and $q$ such that: $$ p_1 \geq p_2 \geq \cdots \geq p_k\text{ and } q_1 \geq q_2 \geq \cdots \geq q_k \geq \cdots \geq q_h \text{ where: } k < ...
1 vote
1 answer
97 views

If $p^k m^2$ is an odd perfect number with special prime $p$, then must $m^2 - p^k = s^2 - t^2$ hold for some $s$ and $t$?

My present question is as is in the title: If $p^k m^2$ is an odd perfect number with special prime $p$, then must $m^2 - p^k = s^2 - t^2$ hold for some $s$ and $t$? It is known that $m^2 - p^k$ is ...
-9 votes
0 answers
181 views

Does RH emanate from a geometric interpretation of the Riemann $\xi$ function? [closed]

The Riemann $\xi$ function is defined in https://fr.m.wikipedia.org/wiki/Fonction_xi_de_Riemann. I realized that the critical line is the perpendicular bisector of the segment $[0,1]$, so that RH can ...
3 votes
2 answers
622 views

Goldbach conjecture and the difference of two primes

The Goldbach conjecure is not yet proved. But, when an even number is represented as a sum of two primes, is there any knwon result about the difference of the two primes? That is, if $2n$ is a sum of ...
3 votes
0 answers
194 views

Number of zeros of the zeta function along horizontal lines

Are there any known results about the number of zeros of the zeta function along horizontal lines of the complex plane? The Riemann hypothesis states that for any such line the number is at most 1, ...

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