Questions tagged [number-theory]

Questions on advanced topics - beyond those in typical introductory courses: higher degree algebraic number and function fields, Diophantine equations, geometry of numbers / lattices, quadratic forms, discontinuous groups and and automorphic forms, Diophantine approximation, transcendental numbers, elliptic curves and arithmetic algebra geometry, exponential and character sums, Zeta and L-functions, multiplicative and additive number theory, etc.

7,559 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
128
votes
0answers
2k views

Does every ring of integers sit inside a ring of integers that has a power basis?

Given a finite extension of the rationals, $K$, we know that $K=\mathbb{Q}[\alpha]$ by the primitive element theorem, so every $x \in K$ has the form $$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n,$$ ...
73
votes
0answers
2k views

Finding primes so that $x^p+y^p=z^p$ is unsolvable in the p-adic units

On my number theory exam yesterday, we had the following interesting problem related to Fermat's last theorem: Suppose $p>2$ is a prime. Show that $x^p+y^p=z^p$ has a solution in $\mathbb{Z}_p^{\...
62
votes
0answers
2k views

A question about divisibility of sum of two consecutive primes

I was curious about the sum of two consecutive primes and after proving that the sum for the odd primes always has at least 3 prime divisors, I came up with this question: Find the least natural ...
51
votes
0answers
2k views

Is OEIS A248049 an integer sequence?

The OEIS sequence A248049 defined by $$ a_n \!=\! (a_{n-1}\!+\!a_{n-2})(a_{n-2}\!+\!a_{n-3})/a_{n-4} \;\text{ with }\; a_0\!=\!2, a_1\!=\!a_2\!=\!a_3\!=\!1.$$ is apparently an integer sequence but I ...
50
votes
0answers
747 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} \...
40
votes
0answers
1k views

Does the average primeness of natural numbers tend to zero?

Note 1: This questions requires some new definitions, namely "continuous primeness" which I have made. Everyone is welcome to improve the definition without altering the spirit of the question. Click ...
34
votes
0answers
880 views

A Nice Problem In Additive Number Theory

$\color{red}{\mathrm{Problem:}}$ $n\geq3$ is a given positive integer, and $a_1 ,a_2, a_3, \ldots ,a_n$ are all given integers that aren't multiples of $n$ and $a_1 + \cdots + a_n$ is also not a ...
34
votes
0answers
772 views

$\forall n\in\mathbb N:n^x\in\mathbb Q$ implies $x\in\mathbb Z$ - elementary proof?

Consider the following two problems: Show that if for some $x\in\mathbb R$ and for each $n\in\mathbb N$ we have $n^x\in\mathbb N$, then $x\in\mathbb N$. Show that if for some $x\in\mathbb R$ ...
33
votes
1answer
714 views

Given a positive integer $t$ does there always exist a natural number $k$ such that $(k!)^2$ is a factor of $(2k-t)!$?

For all natural numbers $k$ the ratio $$ \frac{(2k)!}{(k!)^2}=\binom{2k}k $$ is an integer. From staring at the Pascal triangle long and hard, we know that these ratios grow rather quickly as $k$ ...
33
votes
0answers
902 views

Determinant of a matrix that contains the first $n^2$ primes.

Let $n$ be an integer and $p_1,\ldots,p_{n^2}$ be the first prime numbers. Writing them down in a matrix $$ \left(\begin{matrix} p_1 & p_2 & \cdots & p_n \\ p_{n+1} & p_{n+2} & \...
33
votes
0answers
799 views

Using the Brun Sieve to show very weak approximation to twin prime conjecture

I recently stumbled across MIT OCW for analytic number theory. As a budding number theorist, my ears perked up and I looked through some of the material haphazardly. I don't really know much about ...
32
votes
0answers
549 views

Continued fraction with prime reciprocal entries

We know that the reciprocals of the primes form a divergent series. We also know that a necessary and sufficient condition for a continued fraction to converge is that its entries diverge as a series. ...
31
votes
0answers
925 views

Stronger versions of Wilson's Theorem

Problem Let $c \in \mathbb{N}$ $;$ $\exists$ a prime $p$ for which: $$p^c \mid (p-1)!+1$$ Does $\exists$ $M$ $\in$ $\mathbb{N}$ $;$ $\forall$ $c \geqslant M$ $;$ $\nexists$ $p$ ...
31
votes
0answers
765 views

Are there infinitely many primes of the form $12345678901234567890\dots$

Related to this question, What is the smallest prime number made of sequential number? are there infinitely many primes of the following form (OEIS A057137)? $1, 12, 123, 1234, 12345, 123456, ...
31
votes
0answers
938 views

Algorithm to find primes up to $n$ in $O\left(\frac{n}{\log n}\right)$?

Consider the problem of given an integer $n$, generating a list of the primes not greater than $n$. An optimized version of the Sieve of Eratosthenes can do such task in $O(n)$, while the more modern ...
30
votes
1answer
869 views

$2^n$th decimal place of $\sqrt{2}.$

Someone on Art of Problem Solving claims to know how to calculate the $2^{2020}$th decimal place of $\sqrt{2},$ and will tell us if everyone gives up. Brute force will not work, nor will a BBP style ...
30
votes
0answers
527 views

How many ways can I arrange the numbers $1$ to $N$ with this divisibility condition?

For the numbers $1, \ldots, N$, how many ways can I arrange them such that either: The number at $i$ is evenly divisible by $i$, or $i$ is evenly divisible by the number at $i$. Example: for $N = 2$,...
29
votes
0answers
384 views

Smallest Subset of $\mathbb{R}_{>0}$ Closed under Typical Operations

Let $S$ denote the smallest subset of $\mathbb{R}_{>0}$ which includes $1$, and is closed under addition, multiplication, reciprocation, and the function $x,y \in \mathbb{R}_{>0} \mapsto x^y.$ ...
24
votes
0answers
391 views

Is every finite list of integers coprime to $n$ congruent $\pmod n$ to a list of consecutive primes?

For example the list $(2, 1, 2, 1)$ is congruent $\pmod 3$ to the consecutive primes $(5, 7, 11, 13)$. But how about the list $(1,1,1,1,1,1,1,1,2,3,4,3,2,3,1) \mod 5$? More generally, we are given ...
24
votes
0answers
403 views

Class field theory for $p$-groups. (IV.6, exercise 3 from Neukirch's ANT.)

I will use notation as in a preious question of mine. This question is from Neukirch's book "Algebraic number theory," page 305, exercise 3. Notation for the problem Let $G$ be a profinite $p$-...
24
votes
0answers
998 views

Are there unique solutions for $n=\sum_{j=1}^{g(k)} a_j^k$?

Edward Waring, asks whether for every natural number $n$ there exists an associated positive integer s such that every natural number is the sum of at most $s$ $k$th powers of natural numbers (for ...
23
votes
0answers
644 views

Dividing the whole into a minimal amount of parts to equally distribute it between different groups.

Suppose we have a finite amount of numbers $x_1, x_2, ..., x_n$ ($x_i\in\mathbb{N}$) and an object that should be divided into parts in such a way that it can be without further dividing distributed ...
21
votes
0answers
663 views

Recent developments in the proof of fermat's last theorem

It's been 20 years since fermat's last theorem was proved by Andrew Wiles. Has there been any simplification in proof in the last 20 years? What I do only know is that different proofs of faltings's ...
21
votes
0answers
453 views

**UNSOLVED** Find an integer $\geqslant2$ that is build up out of only $1$'s and $0$'s in base $1,\;\ldots,\;10$.

This riddle bothers me for a few weeks now and I'm starting to worry that I need some $p$-adic Number theory to solve this. I solve most of the riddles in a day, but this one is just annoying to me. I ...
21
votes
0answers
429 views

Does the sum of reciprocals of all prime-prefix-free numbers converge?

Call a positive integer $n$ prime-prefix-free if for all $k \ge 1$, $\lfloor \frac{n}{2^k} \rfloor$ is not an odd prime. (Odd because otherwise the property is trivial, as every integer greater than ...
21
votes
0answers
339 views

Does the category of algebraically closed fields of characteristic $p$ change when $p$ changes?

EDIT I've now posted this question on mathoverflow. It probably makes sense to post answers over there, unless someone prefers posting here. Let $\mathrm{ACF}_p$ denote the category of algebraically ...
21
votes
0answers
608 views

Relationship between intersection and compositum of fields

This issue came up in a number theory lecture today. Let $K$ be a number field and let $L/K$ be an abelian (finite Galois) extension. Then there exists a primitive $m$th root of unity $\zeta_m$ so ...
20
votes
0answers
3k views

Motivation for Weil pairing

The Weil pairing $$e_\phi:E[\phi]\times E'[\hat{\phi}]\to \mu_n$$ for an elliptic curve is defined as follows. Let $\phi:E\to E'$ be an isogeny of degree $n$ and $\hat\phi:E'\to E$ be the dual ...
19
votes
0answers
214 views

Numbers whose decimal digits are the coefficients of its continued fraction form

A curious question recently crossed my mind. And that is can we construct decimal numbers of the form $$\text{"a.bcdefghij..."}$$ where each letter represents a digit $0-9$ (the number may or may ...
19
votes
0answers
2k views

Matrix generated by prime numbers

Let $p$ be the vector of dimension $n^2$ consisting of ordered prime numbers i.e. $p= [ 1 \ 2 \ 3 \ 5 \ 7 \ldots]^T$ and $A$ be the matrix of dimension $n\times{n}$ constructed with this vector by ...
19
votes
0answers
631 views

Conjecture---Identity for Sieve of Eratosthenes collisions.

Let $$\beta(n,k) = \max_{d \leq k}(d|n)$$ $$S(k)= \sum_{n=1}^{k!} \beta(n,k),$$ $\hspace{20mm}$and $$T(k)=\# \{ ~i\cdot j~~\big|_{i=1}^k \big|_{j=1}^{k!} \}$$ Does $$S(k)=T(k)?$$ See OEIS A126959. ...
19
votes
0answers
490 views

When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ for $\Gamma$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least ...
19
votes
1answer
3k 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 $\left(\frac{-bc}{a}\right)=\left(\frac{-ac}{b}\right)=\left(\...
18
votes
0answers
540 views

Can you obtain $\pi$ using elements of $\mathbb{N}$, and finite number of basic arithmetic operations + exponentiation?

Is it possible to obtain $\pi$ from finite amount of operations $\{+,-,\cdot,\div,\wedge\}$ on $\mathbb{N}$ (or $\mathbb{Q}$, the answer will still be the same)? Note that the set of all real numbers ...
18
votes
0answers
763 views

Prime factor of $2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4$

I checked the prime factors of $$2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4 = 2^{2^{2^2}} + 3^{3^{3^3}}$$ with trial division and found non below $8*10^9$ Nevertheless, the given number has still ...
18
votes
1answer
506 views

$\lfloor x^n\rfloor\lfloor y^n\rfloor$ is a perfect square

Let $x,y\ge 1$ be non-integer real numbers such that $\lfloor x^n\rfloor\lfloor y^n\rfloor$ is a perfect square for all natural numbers $n$. Does it follow that $x=y$? From this question we know the ...
17
votes
0answers
330 views

On injectivity of $F_N$

I am interested in the following problem: Let $F_N:\mathbb{N}\to\mathbb{N}$ be the function that maps a natural number $a$ to the least natural number more than or equal to $a$ for which there exists ...
17
votes
0answers
353 views

Four squares such that the difference of any two is a square?

I. This post asks to find $4$ integers $a,b,c,d$ such that the difference between any two is a square. As mentioned by my answer, it is equivalent to finding $3$ squares such that the difference of ...
17
votes
0answers
411 views

Convergence acceleration technique for $\zeta(4)$ (or $\eta(4)$) via creative telescoping?

Question Is it already known whether the $\zeta(4):=\sum_{n=1}^{\infty}1/n^4$ accelerated convergence series $(1)$, proved for instance in [1, Corollaire 5.3], could be obtained by a similar ...
17
votes
0answers
556 views

Riemann zeta function Euler product for primes equivalent to $3$ mod $4$

Question: can $$ \zeta_1(s) = \prod_{p \equiv 3 \pmod{4}} \frac{1}{1 - p^{-s}} $$ be evaluated or written in terms of standard functions? Details: We can write the Riemann zeta function as \begin{...
16
votes
0answers
436 views

Largest consecutive integer using basic operations and optimal digits?

If you are first time reading this, you may want to read the summary section last. Solution summary and questions Sequence values If the allowed operations are $(+,-,\times,\div)$ and parentheses $(...
16
votes
0answers
415 views

Found an recursive identity (involving a continued fraction) for which some simplification is needed.

This is my second question in this forum; as I previously explained it, I am a "hobbyst" mathematician and not a professional one; I apologize by advance if something is wrong in my question. I enjoy ...
16
votes
0answers
262 views

Does every power of two arise as the difference of two primes?

Conjecture: For each $n\in\mathbb N$ there are primes $q<p$ with $p-q=2^n$. Verified for $n\leq 26$: ...
15
votes
0answers
462 views

Is $\lfloor \zeta(-n) \rfloor$ only prime for $n=23$?

I searched for primes of the form of $\lfloor \zeta(-n) \rfloor$, where $n \in \Bbb{N}$, for a range of $n \le 10^4$ on PARI/GP and found $\lfloor \zeta(-n) \rfloor$ is only prime for $n=23$. My ...
14
votes
0answers
389 views

Fibonacci-like sequences mod $p$ where $a_{n+1}$ only really depends on $a_n$.

Consider a prime $p$ and a sequence $(a_n)_{n\ge 0}$ in $\mathbb{F}_p$ satisfying $a_{n+2}=a_{n+1}+a_n$ for all $n\ge 0$. Now, assume that each element of the sequence only really depends on the ...
14
votes
0answers
166 views

Rare gaps between integers having the same number of co-primes

Let $\varphi(x)$ be the Euler totient function. For $k = 1,2,3,\ldots$ I calculated the number of solutions of $\varphi(x) = \varphi(x+k)$. I observed that we have very few solutions when $k = 3,9,15,...
14
votes
0answers
151 views

Is $\Phi(q)$ rational for some $q \in \mathbb{Q}^*$, where $\Phi$ is the standard normal cumulative distribution function?

Suppose that we have rational numbers $q_1$, $q_2$ such that $$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{q_1}e^{-\frac{t^2}{2}} \,\mathrm{d}t=q_2.$$ Does this imply that $q_1=0$ and $q_2=\dfrac{1}{2}$?
14
votes
0answers
368 views

How to prove this $p^{j-\left\lfloor\frac{k}{p}\right\rfloor}\mid c_{j}$

Let $p$ be a prime number and $g\in \mathbb{Z}[x]$. Let $$\binom{x}{k}=\dfrac{x(x-1)(x-2)\cdots(x-k+1)}{k!} \in \mathbb{Q}[x]$$ for every $k \geq 0$. Fix an integer $k$. Write the integer-valued ...
13
votes
0answers
207 views

Are $3^6-6^3$ and $4^8-8^4$ the only sums of four $a^b-b^a,1\lt a\lt b$ numbers?

Question How many numbers of form $a_0^{b_0}-b_0^{a_0}$ are a "nontrivial" sum of four such numbers $a_i^{b_i}-b_i^{a_i}$ ? The "nontrivial" means: all unordered pairs $\{a_i,b_i\}$...
13
votes
1answer
437 views

Is every even number greater than 4 the sum of a prime number with another number that belongs to the set of twin primes?

Thinking about Goldbach conjecture, I have the following question: Is every even number greater than 4 the sum of a prime number with another number that belongs to the set of twin primes? For ...

1
2 3 4 5
152