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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12
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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 ...
1
vote
2answers
133 views

Moving powers in a prime modulus

Suppose I have $$x^{(c(p-1))} \equiv y^{(p-1)} \pmod{p}.$$ I would like to take the (p-1) root of both sides to get: $$x^c \equiv y \pmod{p}$$ I really just want to know if this a valid technique and ...
0
votes
1answer
233 views

Hilbert-Waring theorem using the sum-of-squares function

Denote by $r_{s,k}(x)$, the number of ways in which $x$ can be expressed as the sum of $k$ $s^{th}$ powers of integers. Now, the Hilbert-Waring theorem is equivalent to the following. $$\forall s\in ...
2
votes
0answers
76 views

L-function of product

Given two varieties of finite type over a finite field, what is the $L$-function of their product in terms of the $L$-function of the factors?
1
vote
0answers
135 views

Number theory - Divisibility and perfect square related [duplicate]

Possible Duplicate: Alternative proof that $(a^2+b^2)/(ab+1)$ is a square when it's an integer. I've tried many different approaches and they don't seem to lead anywhere. The question ...
12
votes
1answer
323 views

Primey Pascal's Triangle

Imagine that we have a triangle that starts with 2,3 and grows like Pascal's triangle but instead uses the smallest prime $\geq$ to the sum of the above two primes. Visually: $$ ...
6
votes
1answer
351 views

How to derive an identity between summations of totient and Möbius functions

I have the following identities $$\sum_{n \le x} \varphi(n) = \frac{1}{2} \sum_{n \le x} \mu(n) \left[\frac{x}{n}\right]^2 + \frac{1}{2}$$ $$\sum_{n \le x} \frac{\varphi(n)}{n} = \sum_{n \le x} ...
-3
votes
1answer
301 views

On Albanese varieties

Can anyone give me a precise information or formulation of Birch and Swinnerton-Dyer conjecture for Jacobians -- I mean for Albanese varieties. Any reference to useful links or expository articles, ...
2
votes
1answer
477 views

Four Fours puzzle

The theory is here. It is pretty simple: form any integer bigger or equal that 0 using four fours and symbols. Is there any demonstration which explains why with four fours is possible to form ...
7
votes
1answer
870 views

What is the discriminant of a quadratic extension over a number field?

Let $K$ be a number field and $d \in \mathcal{O}_K \setminus \mathcal{O}_K^2$. What is the discriminant of the extension $K[\sqrt{d}]/K$ ? Do we know its ring of integers and which primes are split or ...
6
votes
4answers
328 views

Prime Appearances in Fibonacci Number Factorizations

Okay, THIS one is considerably more analytical... :P (Used my post here as a basis.) When successive Fibonacci numbers are factored, the primes appear in a specific order, which goes $2, 3, 5, 13, 7, ...
6
votes
2answers
233 views

Injection of $E(\mathbb{Q})_{\text{tors}}$ into $\tilde{E}(\mathbb{F}_p)$?

I'm looking at Example VII.3.3.3 (p.193, 2nd ed.) of Silverman's The Arithmetic of Elliptic Curves. We have the elliptic curve $E:y^2=x^3+x$, with discriminant $\Delta=-64$, so there is good reduction ...
6
votes
2answers
637 views

Asymptotic formula for $\sum_{n\leq x}\mu(n)[x/n]^2$ and the Totient summatory function $\sum_{n\leq x} \phi(n)$

I would like to show (for $x \ge 2$) that $$\sum_{n \le x}\mu(n)\left[\frac{x}{n}\right]^2 = \frac{x^2}{\zeta(2)} + O(x \log(x)).$$ I already have the identity $$\sum_{n \le ...
10
votes
1answer
294 views

Primes of the form 1..1

For $n \ge 1$ an integer, let's denote $u_n = \sum_{k = 0}^{n-1} 10^k$ That is $u_1 = 1$, $u_2 = 11$, $u_3 = 111$, $u_4 = 1111$, ... My question is the following : Which of them are prime numbers ? ...
2
votes
1answer
495 views

Good book resources (not websites) to learn number theory on my own? [duplicate]

Possible Duplicate: Undergraduate/High-School-Olympiad Level Introductory Number Theory Books For Self-Learning I took number theory this semester and loved it but don't feel like I learned ...
7
votes
1answer
168 views

Self-collisions in a stack of cards

(Copied verbatim from: A Project Euler thread I made in December of 2009.) Lets say you have cards numbered from 1 to 5. You want to organize them into a stack such that you can move 1 card from the ...
26
votes
3answers
2k views

Subjects studied in number theory

I know little about number theory even after reading its Wikipedia article. I was wondering if the subjects studied by number theory is only about integers or even more restrictedly natural numbers ...
1
vote
1answer
338 views

Breaking RSA in a special case

This is a part of homework assignment, and I am stuck. The RSA signature is being calculated using Chinese Remainder theorem technique. Find the detailed description here. Public and private keys are ...
6
votes
5answers
656 views

Can this number theory MCQ be solved in 4 minutes?

The Problem: ( 270 + 370 ) is divisible by which number? [ 5, 13, 11 , 7 ] Using Fermat's little theorem it took more than the double of the indicated time limit. But I would like to solve it quickly ...
8
votes
3answers
739 views

On the probability that two positive integers are relatively prime

In many of the sources I have consulted about this, the "probability" that two positive integers chosen at random are relatively prime is calculated as the limit as $n \to \infty$ of the probability ...
12
votes
2answers
692 views

nth powers modulo all primes

Let $a \in \mathbb{Z}$, $n \in \mathbb{N}^*$ be integers, and set $P=X^n - a$. Let us consider the three following statements : 1) $P$ has a root in $\mathbb{Z}$ (i.e. $a$ is an nth power) 2) $P$ ...
4
votes
2answers
225 views

RSA: Creating a key of desired length

Thanks and with respect to the users of this site, I've succeeded in creating an Encryption/Decryption procedure for the RSA algorithm. I also implemented a Miller-Rabin probabilistic primality test. ...
18
votes
3answers
1k views

Prime numbers stretch to infinity, but what about the distance between them?

That is, let $p_n$ be the nth positive prime number. Does $$L = \lim\limits_{n \to \infty} \left( p_{n+1} - p_n \right)$$ equal infinity?
9
votes
1answer
97 views

Does another chain of three squares in this manner exist?

One of my computer science teachers came up with this neat little problem that he and I have both brute-forced and done some analytical work, but we haven't gotten far. There exist chains of squares ...
2
votes
2answers
161 views

sums of reciprocals

Here is another tricky question: Suppose $m_1,\cdots,m_j$ and $m'_1,\cdots,m'_j$ are positive integers such that $\sum\limits_{k=1}^j m_k = \sum\limits_{k=1}^j m'_k$ and ...
13
votes
4answers
471 views

Example of a rational function such that : $(f(x))^{3} + (g(x))^{3} + (h(x))^{3}=x$

Can any one give me example of: rational functions $f, g$ and $h$ with rational coefficients such that $$(f(x))^{3} + (g(x))^{3} + (h(x))^{3}=x$$ Also, if anyone knows a procedure for constructing ...
4
votes
4answers
233 views

Help to find cubics with square discriminant

If the discriminant $b^2-4c$ of the quadratic $x^2 + bx + c$ is a square then it factors. For every discriminant $d^2$ we have can parametrize them all $(b,c) = (d + 2 h,h(d+h))$. edit I realized now ...
0
votes
3answers
153 views

Midpoints of segments joining lattice points

Show that, given $2^n+1$ points with integer coordinates in $\mathbb R^n$, there exists a pair of points among them such that all the co-ordinates of the midpoint of the line segment joining them are ...
2
votes
3answers
940 views

About the factors of the product of prime numbers

If a number is a product of unique prime numbers, are the factors of this number the used unique prime numbers ONLY? Example: 6 = 2 x 3, 15 = 3 x 5. But I don't know for large numbers. I will be using ...
3
votes
1answer
106 views

Exercise on height functions

I'm attempting to solve Exercise VIII.8.8b in Silverman's Arithmetic of Elliptic Curves: Prove that $H(x_1+\cdots+x_N)\leq NH(x_1)\cdots H(x_N)$. This should be elementary, but I'm having ...
-2
votes
1answer
284 views

On Tamagawa number conjecture

In response to my query on an idea which is new idea on B.S.D conjecture ,i got a reply from various people,in that someone told me that proving the Tamagawa number conjecture for eigenforms of weight ...
6
votes
2answers
1k views

What is isogeny?

Can it be explained in a concrete fashion what isogeny is? Apparently some of the proofs of Fermat's last theorem are essentially this but it is not clear to me what that actually is. (It would be ...
11
votes
1answer
1k views

The Sum of Perfect Squares

In Symmetry and the Monster, I ran across this interesting fact: Let $\displaystyle f(n) = 1^2 + 2^2 + 3^2 + \cdots + n^2 = \sum_{k=1}^{n} k^2$ Let $x$ be an integer Then $f(n) = x^2$ for only two ...
3
votes
1answer
147 views

The form $xy+5=a(x+y)$ and its solutions with $x,y$ prime

In another question I was asking if there are any different $x,y>2$ primes such that $xy+5=a(x+y)$. Where $a=2^r-1$, and $r>2$. I was thinking if it is able to find a Pell equation or a ...
0
votes
1answer
263 views

How can I find all increasing sequences $\{a_i\}_{i=1}^{\infty}$ such that $d(x_1+x_2+\cdots+x_k)=d(a_{x_{1}}+a_{x_{2}}+\cdots + a_{x_{k}})$?

How can one find all increasing sequences $\{a_i\}_{i=1}^{\infty}$ such that $$d(x_1+x_2+\cdots+x_k)=d(a_{x_{1}}+a_{x_{2}}+\cdots + a_{x_{k}}),$$ holds for all $k$-tuples $(x_1,x_2,\cdots,x_k)$ of ...
2
votes
1answer
426 views

Finding a cyclotomic field implied by Kronecker-Weber

I was wondering what are the most efficient ways to find a cyclotomic field s.t. given $K/\mathbb{Q}$ we have $K\leq \mathbb{Q}(\zeta_n)$? For quadratic fields this is easy by just considering factors ...
2
votes
0answers
58 views

Exact statement of Artin reciprocity for ray classes

It seems like this particular theorem is always stated in a way that's slightly hard to interpret. Let $S$ be some finite set of primes containing all the primes of $K$ ramifying in $L/K$. Then the ...
4
votes
1answer
173 views

Source of the problem: does there exist k,n>2, such that $\sum_{j=1}^k j^n = (k+1)^n $?

In my other question two days ago I asked for confirmation, whether one step of an attempt to that problem does there exist $k,n>2$ , such that $\sum_{j=1}^k j^n = (k+1)^n $ ? was ...
-2
votes
2answers
375 views

Comment on Heuristic Approach of B.S.D Conjecture

I have read in the history of how Sir Swinnerton-Dyer and Prof. Bryan Birch, have found this conjecture,in that I have found a line like this, ...heuristically the value of the Hasse-Weil ...
5
votes
2answers
116 views

Factorization of L-functions

I'm trying to decipher Lang's Algebraic Number Theory when it comes to L-functions. Let $K/\mathbb{Q}$ be an abelian extension. Then we are supposed to have a factorization ...
7
votes
1answer
123 views

Proof for the uniqueness of all the combination of the digits of $2^k$, where k>3

A long time ago I found a question on the internet that went a little like this: Suppose that we have $n=2^k$ where $k\gt 3$. If $m$ is another number that is a combination of the digits of $2^k$, ...
2
votes
0answers
216 views

Finding out how primes split in a particular cubic extension

I'm trying to figure out as precisely as possible how primes split in a cubic extension $\mathbb{Q}(x)$, where $x^3-x-1=0$. This extension has discriminant $-23$. Write $K=\mathbb{Q}(x)$, ...
13
votes
3answers
759 views

Can multiplication be defined in terms of divisibility?

Peano Arithmetic has two axioms which use the multiplication symbol: ∀x:x*0=0 and ∀x:∀y:x*Sy=x+x*y. The 2-term relation "x divides y" can be expressed as D(x,y) := ∃z:z*x=y. Multiplication is a ...
9
votes
1answer
435 views

Exponential Diophantine Equations for Beginners

What would be some exponential Diophantine equations for the beginner to solve (which can demonstrate the techniques?) especially good if there are hints! Thank you very much!
23
votes
1answer
458 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 ...
4
votes
1answer
103 views

Cokernel of morphism of Tate module of elliptic curves

Let $K$ be a field, and $\phi: E_1\to E_2$ be an isogeny of elliptic curves over $K$. Given a prime $\ell$ different from the characteristic of $K$, $\phi$ induces an injection $T_\ell \phi: T_\ell ...
6
votes
2answers
569 views

Infinite Prime Proof Using Euler's Totient

I need something explained or corrected: In my number theory class we proved that there are an infinite number of primes using Euler's Phi Totient. It went something like this: Let $M = p_1 p_2 ...
7
votes
1answer
86 views

arithmetic/category theoretic information encoded in $q$-series reciprocals

According to the pentagonal number theorem: $$\prod_{n=1}^{\infty} (1-q^{n}) = \sum_{k=-\infty}^{\infty} (-1)^{k}q^{k(3k-1)/2}$$ Now the reciprocal of this has the partition numbers $p(k)$ in its ...
33
votes
1answer
656 views

Geometric intuition behind The Mordell Conjecture

The Mordell Conjecture/Faltings Theorem says roughly that if $K$ is an algebraic number field and $X$ is an algebraic curve defined over $K$ of genus $g >1$ then the set of $K$-rational points ...