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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5
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3answers
79 views

Why $f\colon \mathbb{Z}_n^\times \to \mathbb{Z}_m^\times$ is surjective?

If $m|n$. Why the map $f\colon \mathbb{Z}_n^\times \to \mathbb{Z}_m^\times$ given by $a \mod{n}\mapsto a \mod m$ is a surjective homomorphism of groups? Attempt: I proved it is well a well defined ...
1
vote
2answers
133 views

How to prove $~(c - b) ^ 2 + 3cb = x^3~$ has no nonzero integer solutions?

I'm trying solve: $~a^3 + b^3 = c^3~$ has no nonzero integer solutions. Only one problem left: because $c^3 - b ^ 3 = (c - b)((c - b) ^ 2 + 3cb) = a ^ 3,\quad (1)$ if $~c-b~$ is a cubic number, ...
3
votes
0answers
55 views

Cramér's Model - “The Prime Numbers and Their Distribution” - Part 4

Following a previous question (here you'll find an introduction): A paper by Maier which refutes Cramer's Model suggests we should replace the heuristic "$\Bbb P(n\in\mathcal P)=1/\log n$" with ...
1
vote
0answers
25 views

Cramér's Model - “The Prime Numbers and Their Distribution” - Part 3

Following a previous question (here you'll find an introduction): The book states that almost surely $$\pi_S(x+y)-\pi_S(x)=\mathrm{li}(x+y)-\mathrm{li}(x)+O(\sqrt y)$$ as soon as $y/(\log ...
2
votes
0answers
40 views

Cramér's Model - “The Prime Numbers and Their Distribution” - Part 2

Following a previous question (here you'll find an introduction): The book states that using the convergence of the binomial distribution towards the Poisson distribution, it's easy to show that ...
0
votes
0answers
275 views

Cleaning minimum tables

Moderator Note: This question is part of the Ongoing August Challenge 2014 CodeChef (problem page). This contest ends on 11 August 2014, and this question will remain locked (with current answers ...
7
votes
1answer
51 views

An extension of an algebraic number field which makes an integral ideal $I$, a principal ideal

I want to show that, given an ideal $I \subseteq \mathcal O_K$ (where $K/\mathbb Q$ is an algebraic number field), there is a finite extension $K'/K$ such that, $I\mathcal O_{K'}$ becomes a principal ...
2
votes
2answers
164 views

How to prove $~\sqrt{3}\sqrt{4a^{3}-1}~$ isn't an integer?

I'm trying solve: $~a^3 + b^3 = c^3~$ has no nonzero integer solutions. If $~(c−b)=1~$ then $~c^3-b^3=3c^2-3c+1=a^3,~$ from Wolframalpha get: $$ c = \dfrac{3- \sqrt{3}\sqrt{4a^{3}-1}}{6} \\ c = ...
4
votes
1answer
65 views

Milne's Galois Theory Example

The following example is drawn from Milne's Galois Theory notes, p.42 (http://www.jmilne.org/math/CourseNotes/FT.pdf) We study the extension $\mathbb{Q}[\zeta]/\mathbb{Q}$ where $\zeta=e^{2\pi i/7}.$ ...
0
votes
6answers
76 views

Find 2 square numbers with certain distance

Is there a way to find 2 sqare numbers with a certain distance without trying every square number? Example: $$ a^2 + 204 = b^2 $$
0
votes
0answers
576 views

Count arrangment such that each person wear different tshirt

Few friends are going to a party. Each person has his own collection of T-Shirts. There are 100 different kind of T-Shirts. Each T-Shirt has a unique id between 1 and 100. No person has two T-Shirts ...
4
votes
3answers
88 views

Does there exist a cubic polynomial $f(x)$ such that $f(x)\equiv0 \pmod p $ has no integer solutions if $p\equiv 3\pmod 4$?

As we know that $f(x)=x^2+1\equiv0 \pmod p $ has no integer solutions if $p\equiv 3\pmod 4$, does there exist a cubic polynomial $f(x)=ax^3+bx^2+cx+d~(a,b,c,d \in\mathbb Z,a\neq 0) $ such that ...
0
votes
0answers
35 views

What is the explanation for the $64$ in Graham's number $g_{64}$?

As in, why does the iteration of the function until $g_{64}$ guarantee this property that defines Graham's number? Why was this number chosen? If I had to guess (emphasis on guess), I'd say that the ...
0
votes
1answer
30 views

Dedekind rings which are UFDs but not PIDs?

I just have a really quick question of an example that I was trying to come up with. Are there any number rings which are UFDs but not PIDs?
2
votes
1answer
48 views

Computing $\text{Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$

Algebraic class field theory tells us that $\text{Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$ is isomorphic to the group of connected components of the quotient $\mathbb{Q}^{\times}\backslash ...
0
votes
3answers
184 views

1729, and related questions

I just read this paragraph: (written by G. H. Hardy, on Ramanujan) I remember once going to see him when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the ...
2
votes
1answer
67 views

Finding an $n$ such that $n^2 \equiv -1 \mod p$

What is an efficient algorithm to find the first number $n$ such that $n^2 \equiv -1 \mod p$ for a prime $p$, if such an $n$ exists? Is there anything better than the brute-force approach up to $p-1 ...
3
votes
1answer
52 views

Nice polynomial reducibility: $x^n+4$

Problem: Find all $n\in \mathbb{N}$ such that $f(x)=x^n+4$ is reducible in $\mathbb{Z}[x]$. It seems $n=4k$ is the only one (the factorization follows easily from Sophie Germain's identity in this ...
6
votes
1answer
53 views

Torsion of finitely generated $\mathbb Z_ \ell$-module finite?

Let $M$ be a finitely generated $\mathbb Z_ \ell$-module, where $\ell$ is a prime number and $\mathbb Z_\ell$ is the ring of $\ell$-adic integers. Let $T$ be its torsion submodule. Is $T$ finite? In ...
3
votes
1answer
60 views

Calculate 'interference' of number patterns

I have 2 numerical series like this: $$ 144 + 25 + 27 + 29 + 31 + \cdots $$ $$ 133 + 3 + 5 + 7 +9 +11+13+\cdots $$ Is there a efficient way to find the common sum of these patterns? solution for ...
2
votes
1answer
46 views

Unique Products on a Times Table

I was looking at a 10x10 multiplication table, and I decided to count the unique products. There are 42 out of a possible 100 numbers represented. I had to wonder, why 42? I counted the 58 non-listed ...
0
votes
0answers
15 views

Negative Weight meromorphic modular forms/ Sections of Line bundles

it is known, that we can see modular forms as section of line bundles on a Riemann surface. Especially, we know that a meromorphic modular form of weight 2 on SL(2,Z) corresponds to a meromorphic ...
4
votes
1answer
146 views

Squeezing $\pi(x)$ out of $\psi(x)$

Can $\pi(x)$ be written in terms of $\psi(x)$? I can only seem to approximate it: $$ ...
-1
votes
0answers
37 views

Sparsity of Linear Diophantine Equations

If you are looking for integer solutions to the system. $$Ax=b$$ where $A$ is an integer matrix and $b$ is integer vector, then you can construct the solution space integer matrix $B$ and integer ...
1
vote
2answers
43 views

Can't understand source of constant for prime counting function:

Consider the prime counting function $$ \pi(x) = \ the \ number \ of \ primes \ less \ than \ or \ equal \ to \ x$$ It is well known due to the sieve eratosthenes that given an integer $n$ and the ...
1
vote
4answers
149 views

Why didn't Fermat provide proofs of his theorems?

Apparently Fermat stated but didn't provide proofs of various theorems named after him, including Fermat's little theorem, Fermat's theorem on sums of two squares, Fertmat's polygonal number theorem, ...
1
vote
1answer
64 views

Is $\{1,1,2,3,4,5,\cdots,i,\cdots \} $ the simple continued fraction algebraic or transcendental?

Is $$1+\cfrac{1}{1+\cfrac{1}{2+\cdots}} $$ or$\{1,1,2,3,4,5,\cdots,i,\cdots \} , i\in \mathbb{N}$ the simple continued fraction algebraic or transcendental? Any reference is appreciated EDIT and ...
2
votes
2answers
73 views

Diophantine equation: $n^p+3^p=k^2$

Find all solutions to the Diophantine equation $n^p+3^p=k^2$, where $p\in \mathbb{P}$ and $n,k$ positive integers. I have tried everything, from mods to bounding to LTE; nothing seems to work on ...
3
votes
1answer
42 views

Having trouble with binary quadratic forms.

A quadratic form represents an integer $n$ if there exist $x,y\in \mathbb{Z}$ such that $f(x,y)=n$. It is proper if $\gcd{(x,y)}=1$. It is said that if $f(x,y)=n$ and $\gcd{(x,y)}=g$, then $g^2|n$. ...
0
votes
1answer
20 views

Any upper bound for $a_i$ in $\gamma =\{a_0,a_1,\dots,a_i,\dots\}$ the simple continued fraction expansion of real positive algebraic numbers?

Are there any upper bound for $a_i$ in $\gamma =\{a_0,a_1,\dots,a_i,\dots\}$ the simple continued fraction expansion of real positive algebraic numbers?
0
votes
0answers
44 views

Infinitely many prime numbers 6n-1 [duplicate]

Prove that there are infinitely many prime numbers of the form $6n-1$. I proved that there are infinitely many prime numbers but I couldn't bring it in the form given in the question. While proving ...
2
votes
0answers
39 views

What's the proof for the #integers less than $n$ that can be expressed as the sum of two squares is $\frac n{\sqrt{\log n}}$?

This result is used in the Erdos' Distance problem, in the Landau-Ramanujan constant, but I can't find a proof anywhere. http://en.wikipedia.org/wiki/Erd%C5%91s_distinct_distances_problem ...
2
votes
1answer
48 views

Algebraic Integers in $\mathbb{Q}(\sqrt{m})$ and Norms on them

I'm having a problem with a section of Niven's book the Theory Of Numbers. I am trying to show: If an integer $\alpha \in \mathbb{Q}(\sqrt{m})$ is neither zero nor a unit, prove that ...
1
vote
1answer
26 views

Expanding Base 2B representation of an integer

Consider an integer$L$ written in Base 2B which digits $$a_n a_{n-1} a_{n-2} ... a_1 B$$ Where $a_i$ are arbitrary constants such that $9 \le a_i < 2B$. I am attempting to prove that the square ...
1
vote
0answers
33 views

A Question Related to Zsigmondy's Theorem

I am wondering if there is a way to prove the following statement, which bears some resemblance to Zsigmondy's Theorem. I am not sure if the statement is true, but it seems as though it should be. ...
-2
votes
2answers
310 views

Is this a solution for the problem: $\ a^3 + b^3 = c^3\ $ has no nonzero integer solutions?

Is this a solution for the problem: $\ a^3 + b^3 = c^3\ $ has no nonzero integer solutions? Suppose $\ a^3 + b^3 = c^3,\ a,b,c \in \mathbb Z^*,\ $then: $c^3 - b ^ 3 = (c - b)((c - b) ^ 2 + 3cb) = ...
3
votes
1answer
37 views

Non unique factorization domains with prime factorizations with differing number of primes

As is well-known, $Z[\sqrt{-5}]$ is not a ufd because $6$ has more than one prime factorization in this ring: $6=2\cdot 3$ and $6=(1+\sqrt{-5})(1-\sqrt{-5})$. But both of these prime factorizations ...
3
votes
0answers
46 views

Are there any primes that are never a factor of a Carmichael number?

Is there a prime number $p$ that $p > 2$, and in which $p$ is a never a factor of any Carmichael number $C_n$: (p ∤ $C_n$) Extended this to all numbers $m$, instead of just $p$, will prove the ...
1
vote
3answers
33 views

Ideals of the residual classes $\mathbb Z_n$

Let $n$ be a positive integer and considere the ring $\mathbb Z_n$ of the residual classes modulo $n$. My intuition tells me that there is no two distinct ideals of $\mathbb Z_n$ with the same number ...
3
votes
2answers
64 views

Is 7 prime or irreducible or something else in $\mathbb{Z}_{21}$

I thought I understood prime numbers pretty well, but now I'm told about this thing called irreducible, that sometimes numbers are irreducible but not prime (like 3 in $\mathbb{Z}[\sqrt{-5}]$) and ...
3
votes
3answers
115 views

Does $x^2+x+1 \equiv 0 \pmod {997}$ have solutions? Why or why not?

I'm have difficulty solving this problem in my textbook. Does $x^2+x+1 \equiv 0\pmod{997}$ have solutions? Why or why not? I guess the first step would be $$ \begin{array}{l} (2x+1)^2 \equiv ...
1
vote
0answers
27 views

Asymptotic behavior of sums of consecutive powers (bivariate)

Are there some (bivariate) closed form formulas for the asymptotic behaviour of the sum: $$\sum_{k=1}^{n} k^d,$$ where $n$ and $d$ are large integers? I am especially interested in a lower bound of ...
0
votes
2answers
19 views

Scan through all integers within a range by incrementing constant amount

Given a natural number $k$, what is a way to find out all natural number $i$ such that, when we start with $n = 0$ and keep adding $i$ to $n$, the value $n \mod k$ traverses through all numbers ...
1
vote
1answer
61 views

Equivalent definitions of a lattice in a real vector space of finite dimension

I'm currently trying to work my way through chapter seven of Serre's book "A Course in Arithmetic" with a view to learning about modular forms. During the course of this chapter the book begins to ...
3
votes
1answer
53 views

An elementary question regarding a multiplicative character over finite fields

Reading Chapter 2 of Koblitz's Introduction to Elliptic Curves and Modular Forms, I got stuck on the following question. I would like to proceed my reading, so I would appreciate any hint to this. I ...
2
votes
3answers
58 views

Diophantine Equatiοn $x^3=2^y+15$

I would like some help with the diophantine equation $x^3=2^y+15$ I have tried working with last digits and modular arithmetic but that hasn't got me anywhere.
0
votes
2answers
36 views

Let $p$ be an odd prime and $(a,p)=1$. Show that $x^2≡a$ (mod $p)$ has solutions, then $x^2≡a$ (mod $p^n)$ always has solution , for any $n>1$.

Let $p$ be an odd prime and $(a,p)=1$. Show that $x^2≡a$ (mod $p)$ has solutions, then $x^2≡a$ (mod $p^n)$ always has solution , for any $n>1$. I have solved the first part but second part need ...
0
votes
2answers
54 views

Let $p > 3$ be a prime number. Show that $x^2 \equiv −3\mod p$ is solvable iff $p\equiv 1\mod 6$.

Let $p > 3$ be a prime number. Show that $x^2 \equiv −3\mod p$ is solvable iff $p\equiv 1\mod 6$. My try is let $a$ be a solution of $x^2 \equiv -3 \mod p$. so $a^{p-1} \equiv 1\mod p$. This ...
2
votes
1answer
44 views

Primitive roots and quadratic nonresidues modulo a prime of form $2^n+1$ [duplicate]

Let $p$ be a prime number. We call a unit $a$ in $\Bbb Z/p\Bbb Z$ a primitive root, if $\text{ord}_p(a)=p-1$. Any unit in $\Bbb Z/p\Bbb Z$ can be written as some power as some power of $a$. if $p$ is ...
1
vote
1answer
18 views

Evaluate the Legendre symbols (503/773) and (501/773)

Evaluate the Legendre symbols (503/773) and (501/773) my solution (501/773 ) = (((167*3))/773 ) = (167/773) * (3/773) = (773/167)*(773/3) = (105/167) * (2/3) = (3/167) * (5/167) * (7/167) * ...