Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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2
votes
1answer
100 views

What is the smallest prime so that an instance of the discrete log problem has a solution?

The decision version of the discrete log problem in $\mathbb{Z} /q \mathbb{Z}$ is to determine whether there exists a k so that $x^k$=y mod q. Let gcd(x,y)=1. I am interested in an upper bound for ...
5
votes
2answers
147 views

Avoiding matching first digit of $a^n$ with $b^n$

For any given pairs of positive integers $a$ and $b$, is it possible that the first digit of $a^n$ never matches the first digit of $b^n$ for any positive integer $n$? (If $a=2$ and $b=5$ the only ...
4
votes
3answers
386 views

Generalization of Dirichlet's theorem

Is there a generalization of Dirichlet's theorem along these lines? If $p(n)$ is a polynomial of degree $k$ with positive integers as coefficients, such that the coefficients are relatively prime, ...
19
votes
7answers
2k views

Mathematical Career Advice to a young 16 year wannabe mathematician

I am a '94 born aspiring mathematician, programmer and a little bit of drummer from Nepal. I have been an avid follower and lover of mathematics since my seventh grade. Although I do not even consider ...
6
votes
0answers
318 views

On the equation $m^3-m^2+1 = n^2$

(i) How can I find all positive integers $m$ such that $m\equiv 4 \pmod 7$ and $m^3-m^2+1$ is a perfect square? (ii) Is there a method to solve this equation over positive integers: $$m^3-m^2+1 = ...
13
votes
3answers
3k views

Analytic continuation- Easy explanation?

Today, as I was flipping through my copy of Higher Algebra by Barnard and Child, I came across a theorem which said, The series $$ 1+\frac{1}{2^p} +\frac{1}{3^p}+...$$ diverges for $p\leq 1$ and ...
0
votes
0answers
203 views

Could anyone please give me some hint on relation between number theory and algebraic geometry,with solid examples please!

I studied math all by my self for a few years out of my own love for it, since without guide, I was often puzzled. I'd like ask for aid here, mainly about why we view numbers over spec Z etc. Does it ...
5
votes
0answers
349 views

Is this a relation between the Riemann zeta function and the Prime zeta function?

I am partly repeating myself here again. Is this a correct relation between the Riemann zeta function and the Prime zeta function? $$ \zeta (s) = \sum\limits_{n=1}^{\infty}\frac{1}{n^{s}}$$ ...
21
votes
4answers
680 views

Is an integer uniquely determined by its multiplicative order mod every prime

Let $x$ and $y$ be nonzero integers and $\mathrm{ord}_p(w)$ be the multiplicative order of $w$ in $ \mathbb{Z} /p \mathbb{Z} $. If $\mathrm{ord}_p(x) = \mathrm{ord}_p(y)$ for all primes (Edit: not ...
0
votes
1answer
134 views

Completion of algebraic numbers?

The algebraic closure of the field of real numbers is the field of complex numbers. The algebraic closure of the field of rational numbers is the field of algebraic numbers. The completion of the set ...
9
votes
1answer
166 views

What is the sum of the squares of the differences of consecutive element of a Farey Sequence

A Farey sequence of order $n$ is a list of the rational numbers between 0 and 1 inclusive whose denominator is less than or equal to $n$. For example $F_6= ...
7
votes
1answer
167 views

Transcendentality of the $\log$ of the golden mean

We know that $\phi$, the golden ratio, is algebraic. Is it known whether $\log(\phi)$ is algebraic? Thank you! PS. I am not in number theory, so I apologize in advance if this is obvious.
3
votes
1answer
93 views

Weird infimum of fractional part expression

I found this problem a while ago on AoPS. Let $k$ be a squarefree positive integer. Find $\inf_{n \in \Bbb{Z}_+^*} n\{n\sqrt{k}\,\}$, where $\{\cdot\}$ denotes the fractional part. I ...
4
votes
1answer
126 views

Primes for which $x^k\equiv n\pmod p$ is solvable: the fixed version

For fixed $n$ and $k$, how can I characterize the primes $p$ such that $x^k\equiv n\pmod p$? Less important to me: Is there a similar characterization for composite moduli? Assume the factorization ...
14
votes
8answers
13k views

Efficient way to determine if a number is Perfect Square

Is there an efficient method to determine if a very large number (300 - 600 digits) is a perfect square without finding its square root. I tried finding the square root but I realized that even for ...
3
votes
2answers
147 views

Primes for which $x^k \equiv n \pmod{p}$ is solvable

For a fixed $n$, how can I characterize the primes $p$ such that there is a $k$ with $x^k\equiv n\pmod p$? Edit: This wasn't actually what I meant... the question I intended is here.
1
vote
1answer
183 views

Product of odd integers in Mod M

Does anyone know of any formular or algorithm (that runs sub-linear time) for computing the product of all odd integers in an interval in (mod M), or a similar product?
3
votes
2answers
190 views

Fibonacci/Lucas Number Congruences

Is there a compendium of well-known (and elementary) Fibonacci/Lucas Number congruences? I've proven the following and would like to know if it is (a) trivial, (b) well-known, or (c) possibly new. $$ ...
14
votes
1answer
386 views

Fractional Part Double Summations

In attempt to deepen my understanding of Dedekind sums, I've proven the following identity $$ \sum_{i = 0}^{t} \sum_{j = 0}^{b(t-i)} \left \lbrace c \left( t - i - \frac{j}{b} \right) \right \rbrace = ...
11
votes
1answer
598 views

What is the inverse of the Carmichael-function?

Given an integer $y$, how can I find the biggest $x$, such that $\lambda(x)=y$, where $\lambda(x)$ is the Carmichael-function?
2
votes
2answers
188 views

A question about integral quadratic forms

Hi Would you please advise me? Consider the equation below: $$ ax^2+bxy+cy^2=n $$ in which $a, b, c$ and $n$ are integers. We then suppose that $a, b, c$ are constant. Is there any way to find the ...
37
votes
3answers
1k views

Triangular Factorials

I came across a statement online and have been looking for a proof : It states that 1, 6 and 120 are the only numbers which are both triangular and factorials. Is there any way I can prove this? ...
11
votes
1answer
274 views

A quadratic reciprocity formula

Inspired by a problem of calculating explicitly the invariants by Reshetikhin and Turaev for certain 3-manifolds, I have come across the following problem involving Gauss sums: I would like to prove ...
26
votes
2answers
1k views

Why do some Fibonacci numbers appear in an approximation for $e^{\pi\sqrt{163}}$?

It is rather well-known that, $e^{\pi\sqrt{43}} \approx 960^3 + 743.999\ldots$ $e^{\pi\sqrt{67}} \approx 5280^3 + 743.99999\ldots$ $e^{\pi\sqrt{163}} \approx 640320^3 + 743.999999999999\ldots$ Not ...
1
vote
2answers
117 views

Two succeeding integers in $\left(\mathbb{Z}/n\mathbb{Z}\right)^{*}$ for an odd n, and the Jacobi symbol of the latter one

Given an odd integer $n$, I want to find out if there exists two succeeding integers, $1\leq m-1<m\leq n-1$ s.t both are invertible (i.e $m,m-1\in \left(\mathbb{Z}/n\mathbb{Z}\right)^{*}$) and also ...
5
votes
1answer
141 views

Diophantine equation : $N= \frac{x^2+y}{x+y^2}$

I am looking for information about the following diophantine equation : $N = \displaystyle\frac{x^2+y}{x+y^2}$ Has it been studied ? Is there any efficient algorithm to solve it? Any links? I ...
3
votes
1answer
128 views

Find a finite extension of $\mathbb{Q}$ in which all primes split

Dear all, I would be grateful if someone could provide a solution to the following problem (using decomposition and inertia groups): Find a finite extension of $\mathbb{Q}$ in which all primes split. ...
7
votes
0answers
147 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 ...
1
vote
1answer
242 views

on the application of FLT

i saw a problem which is a generalization of Beals conjecture,here sinha conjecture,it holds a open prize of US$150,000, for the proof or disproof of a mathematical proposition . The award money was ...
0
votes
3answers
1k views

Extract a Pattern of Iterated continued fractions from convergents

I have been working on an article at https://oeis.org/wiki/Table_of_convergents_constants where I posted a table of "convergents constants" (defined at https://oeis.org/wiki/Convergents_constant) ...
11
votes
1answer
372 views

Fibonacci, compositions, history

There are three basic families of restricted compositions (ordered partitions) that are enumerated by the Fibonacci numbers (with offsets): a) compositions with parts from the set {1,2} (e.g., 2+2 = ...
7
votes
2answers
231 views

Given $2$ randomly chosen integers $x,y$ what is $P(k=gcd(x,y))$?

Given $2$ randomly chosen integers $x,y$ what is the probability that a integer $k$ is the greatest common divisor of $x$ and $y$? I know that the probability of $x,y$ being coprime is ...
5
votes
1answer
233 views

Universal quadratic forms

A quadratic form is a polynomial $p(x_1,\dots,x_n)$ of the form $$ p(x_1,\dots,x_n)=\sum_{i \leq j}a_{ij}x_ix_j. $$ For example, $p_1(x,y,z,w)=x^2+y^2+z^2+w^2$ and $3x^2-5y^2$ are quadratic forms. I'm ...
5
votes
4answers
436 views

How to compute next/previous representable rational number?

An (approximate) non-negative rational number representation is a pair of natural numbers each not greater than some fixed limit M (and of course denominator being non-zero). With this condition ...
20
votes
1answer
435 views

Derivative of numbers

I am not talking here about deriving a constant in the usual sense (which gives $0$ as everyone knows). I have once heard of a derivative operator with relaxed conditions that was defined over the ...
3
votes
1answer
151 views

Miller primality test bound

Good morning! I'm on my way to implement a deterministic (though unproven due to GRH) Miller primality test. On Wikipedia, it is said that it suffices to test all numbers in $[2, ...
1
vote
3answers
117 views

Find the number of quadratic polynomials ax² + bx + c

Find the number of quadratic polynomials ax² + bx + c such that: a) a, b, c are distinct. b) a, b, c ε {1, 2, 3, …2008} c) x + 1 divides ax² + bx + c Reasoning->b=a+c ...
5
votes
1answer
174 views

Concerning: presentations of rational numbers into sums

Problem: Prove that all positive rational numbers can be expressed as the finite sum of different numbers $\displaystyle \frac {1} {n}$ ($n$ is a natural number). Example: $\displaystyle \frac ...
3
votes
2answers
1k views

Application of Galois theory

i have a question regarding roots of equation, find all $a$,such that the cubic polynomial $x^3-bx+a=0$ has three integer roots, how can you solve these by using galois theory,what does the reducible ...
6
votes
1answer
96 views

Literature Request: Twisted Dirichlet Series

Recently, I have been pushed toward studying analytic continuation dirichlet series with twists that are additive. These are functions $D(s) = \sum e_k(hn)\frac{a_n}{n^s}$ where $a_n$ is some ...
16
votes
3answers
901 views

Motivation behind this eccentric Ramanujan Identity

I just visited the MathJaX page due to the Math.SE website showing some problems while loading the page. I saw some demo math equations samples at this page, when this identity actually caught my ...
-1
votes
1answer
179 views

Finite extension of $\mathbb Q_p$

Let $\mathbb K/\mathbb Q_p$ be a finite extension of $p-$adic field $\mathbb Q_p$. Let ${\mathcal O}=\{x\in K\;:\;|x|\leq1\}$ and ${\mathcal P}=\{x\in K:\;|x|<1\}$, here $|\cdot|$ is the absolute ...
15
votes
6answers
3k views

Intuitive understanding of why the sum of nth roots of unity is $0$

Wikipedia says that it is intuitively obvious that the sum of $n$th roots of unity is $0$. To me it seems more obvious when considering the fact that $\displaystyle ...
5
votes
3answers
293 views

Analogies between Prime Ideals and Knots

While reading this question posted at this link: Subjects studied in number theory i interestingly landed up on this Wikipedia page, and was quite amazed to see the variety of branches opening up. ...
10
votes
3answers
333 views

Are there any Combinatoric proofs of Bertrand's postulate?

I feel like there must exist a combinatoric proof of a theorem like: There is a prime between $n$ and $2n$, or $p$ and $p^2$ or anything similar to this stronger than there is a prime between $p$ and ...
12
votes
3answers
352 views

Complex Galois Representations are Finite

In A First Course in Modular Forms, Diamond and Shurman leave as an exercise ($9.3.3$) that every complex Galois representation is finite. While I think I have worked through this exercise here, this ...
1
vote
0answers
62 views

Can Fermats descent be interpreted on a conic?

Fermat proved the Diophantine equation $$(x^2)^2 + (y^2)^2 = z^2$$ has only solutions $(0,0,0)$, $(0,\pm 1,\pm 1)$ and $(\pm 1,0,\pm 1)$ using "infinite descent". The conic $C : X^2 + Y^2 - 1$ has a ...
2
votes
0answers
537 views

Graduate studies in Number theory

I'm finishing my undergraduate studies in mathematics and I'd like doing research in a future, probably on Number Theory. That's why I'm interested in graduate program where I could focus on NT. It ...
12
votes
2answers
558 views

When do Pell equation results imply applicability of the “Vieta jumping”-method to a given conic?

This question is motivated by a remark of Bill Dubuque on my answer to the following question: An Algorithm to compute the GCD of polynomials of coprime numbers? Vieta jumping is based on the idea ...
1
vote
2answers
149 views

Another prime inquiry: how many primes from 1 to *k*?

With the question I made about primes, I noticed people enjoy the subject, so here's another thought: let k be a positive integer; how many primes are there from 1 to k? There's probably no exact ...