Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc.. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

learn more… | top users | synonyms (1)

-1
votes
0answers
10 views

Solving $a^2$ $+$ $ b^2$ $=$ $2c^2$

I was working through some number theory problems , when I came across the following question : Find all solutions of $a^2$ $+$ $b^2$ $=$ $2c^2$ Can someone help me out ? Maybe a hint ...
3
votes
3answers
18 views

Understanding the trivial primality test

I'm reading an algorithms book and I came across a code example for primality test. The problem is that I couldn't understand the condition for the for-loop; ...
0
votes
0answers
19 views

General way to solve congruence problems of the following type?

General way to solve congruence problems of the following type? $x^n \equiv a \pmod b$, where $n,a,b$ are constants, $x$ a variable. Also, is this always solvable?
5
votes
2answers
63 views

Irrational Numbers : Show that $0.1248163264…$ is irrational

I was working through some basic Number Theory Problems in Rosen and came across the following problem : Show that the real number $0.1248163264...$ represented in ...
3
votes
3answers
25 views

Largest subset with no arithmetic progression

I am trying to find some weak bounds on the largest subset of a set, such that the subset has the property that it contains no three elements in arithmetic progression. The elements of the original ...
5
votes
3answers
49 views

For any 7 different real numbers, there are among them two numbers x and y such that $0<\frac{x-y}{1+xy} < √3$

For any 7 different real numbers, there are among them two numbers x and y such that $\frac{x-y}{1+xy}$ is greater than zeron and less than the square root of 3. I find this fact quite amazing for ...
4
votes
4answers
115 views

How to solve $x^3\equiv 10 \pmod{990}$? [on hold]

How to solve $x^3\equiv 10\pmod{990}$? It has 3 solutions: 10, 340, 670. Here is the link: https://www.wolframalpha.com/input/?i=x%5E3+%3D+10+%28mod+990%29
0
votes
1answer
55 views

Prove that there are infinitely many primes in $\mathbf Q[\sqrt{d}]$.

Prove that there are infinitely many primes in $\mathbb Z[\sqrt{d}]$. I don't know how to prove this, but I think that the proof will be similar to proving that there are infinitely many primes in ...
-2
votes
1answer
61 views

Is there an arbitray large prime equal to 3k+2? [duplicate]

How could I find an arbitray large prime number equal to 3k+2?
-4
votes
1answer
36 views

If two integers are associates then their norms are equal in absolute value [on hold]

In $Q[\sqrt{d}]$ prove that if two integers are associates then their norms are equal in absolute value when $d>0$ and $d<0$
2
votes
1answer
17 views

Infiniteness of set of primes such $f$ have root $\mod p$ [duplicate]

Let $f \in \mathbb{Z}[x]$ be non constant. How to prove that exists infinitely many primes such $f$ have root in $\mathbb{Z/_{(p)}}$. I spent much time, but with no benefits.
-3
votes
0answers
21 views

define a notion of congruence [on hold]

So I am learning congruences and ring theory etc, and I have a question. If $α$ is a quadratic integer in $Q[√−d]$, then what would define a notion of congruence (meaning mod $α$). Also, how would ...
0
votes
2answers
40 views

Sum of odd Fibonacci Numbers

Trying to prove that the sum of odd-index consecutive Fibonacci numbers is the next even-index Fibonacci number. I have a gap in my proof that I cannot figure out. I know that induction would be ...
-1
votes
0answers
35 views

Choosing M cards from N decks

Alice and Bob are playing cards. They have N decks of cards. Each deck of cards contain 52 cards: ...
1
vote
0answers
8 views

norm map and local class field theory

Let $K$ be a local field, say a finite extension of $\mathbb{Q}_p$ (which is the purpose of my interest). Let $L$ be an unramified extension of $K$. Local class field theory asserts that there ...
-1
votes
2answers
22 views

Congruence Classes $(\text{mod} (3 + \sqrt{−3})/2)$ in $Q[\sqrt{−3}]$ [on hold]

What would be the congruence classes $(\text{mod} (3 + \sqrt{−3})/2)$ in $Q[\sqrt{−3}]$?
0
votes
3answers
40 views

Number of times $2^k$ appears in factorial

For what $n$ does: $2^n | 19!18!...1!$? I checked how many times $2^1$ appears: It appears in, $2!, 3!, 4!... 19!$ meaning, $2^{18}$ I checked how many times $2^2 = 4$ appears: It appears in, ...
3
votes
1answer
45 views

Find all odd positive integers $n$ greater than $1$ such that for any coprime divisors …

Find all odd positive integers $n$ greater than $1$ such that for any coprime divisors $a$ and $b$ of $n$, the number $a + b − 1$ is also a divisor of $n$. This was taken from the Russian ...
1
vote
3answers
41 views

How to apply Chinese Remainder Theorem for $x$

If: $$x \equiv 0 \pmod{17}$$ and $$x \equiv -1 \pmod{9}$$ Then how is: $$x \equiv 17 \pmod{153}$$ I get that since $\gcd(9, 17) = 153 $ the solution will be $\pmod{153}$ but how do you get the $17 ...
0
votes
1answer
11 views

Ideal factorization Theorem, more generally

Consider Theorem 4.3.1 in link (it's quite long, so please open the pdf) I'm wondering if we can assume that the prime ideal we want to decompose is not $(p)$ with $p$ a prime in $\mathbb Q$, but a ...
5
votes
5answers
69 views

solutions such that a combination number is odd

Let $m$ be a positive integer. Given $m$, I want to find the largest $n$, $1\leq n\leq m$, such that $$m+n\choose n $$ is odd. Is there any standard theorems or results? Any references? Thanks!
1
vote
1answer
23 views

Number of $q$-th residues modulo $n$

Let $q$ be a prime and $n\ge 2$ an integer. Moreover, define $f_q(n)$ as the number of $q$-th residues modulo $n$. Is it true that if $K$ is a positive constant then there exist infinitely many $n$ ...
0
votes
0answers
14 views

Jacobians and ranks of a curve

I would like to know the following: How to find Jacobian and rank of an hyper elliptic curve like $x^5-x= y^2-y$? High regards Rosy
11
votes
0answers
25 views

Analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$.

We consider an analogue of the Dirichlet $L$-function in $\mathbb{F}_q[T]$. Let $g \in \mathbb{F}_q[T]$, $g \neq 0$, let $\chi: (\mathbb{F}_q[T]/(g))^\times \to \mathbb{C}^\times$ be a homomorphism, ...
-1
votes
1answer
27 views

Prove that there is an integer a such that a is a primitive root modulo p^2 and a is relatively prime to n. [Hint: Use the Chinese Remainder Theorem.] [on hold]

let n be a natural number, let p be a prime, and suppose $p^2 \mid n$. Prove that there is an integer a such that a is a primitive root modulo $p^2$ and a is relatively prime to n. [Hint: Use the ...
2
votes
1answer
55 views

Most Common Difference Between Two Consecutive Primes?

The question is as stated in the title. I was given this interesting problem by a friend of mine, but I don't know how to proceed with a solution. The immediate thought I had was that the most common ...
0
votes
1answer
66 views

Find the last digit of $(\sqrt{71}+1)^{71}+(\sqrt{71}-1)^{71}$

While teaching binomial expansion, one of my high school students asked me the following question: What is the last digit of $(1+\sqrt{71})^{71}+(1-\sqrt{71})^{71}$? I have absolutely no context on ...
1
vote
1answer
17 views

The norm of Gaussian integers and the irreducible element $ 1 + i $.

Note: Let $ \text{N}(a + bi) \stackrel{\text{df}}{=} a^{2} + b^{2} $. Observe that $ \text{N}(1 + i) = 2 $. Is it always true that if $ 1 + i $ divides a Gaussian integer, then the norm of $ 1 + i $ ...
1
vote
2answers
38 views

If $n \in \mathbb N$, under what conditions are $n$ and $n+2$ relatively prime?

The Statement of the Problem: If $n \in \mathbb N$, under what conditions are $n$ and $n+2$ relatively prime? My Thoughts: I know that the answer is that $n$ must be odd. However, I'm not sure how ...
9
votes
2answers
52 views

Definition of $\mathfrak{m}$-adic completion.

Let $V$ be a discrete valuation ring with the maximal ideal $\mathfrak{m}$ and let $T$ be a prime element of $V$. Assume that we have a subfield $k\subseteq V$ such that the induced map $ k \to ...
1
vote
2answers
53 views

Interesting $0, 1$ sequence of numbers,after $n>2, a_n$is composite.

Let us have a finite sequence with only $0$ and $1$ digit in our numbers(it can begin with $0$ too). $a_n$ is the number, which we get if we write our number $n$ times next to each other. Prove, that ...
1
vote
4answers
53 views

How to come up with proofs of these results? Or, are these results true in the first place?

Let $x_n$ and $y_n$ be integer sequences determined by $$x_n + y_n \sqrt{2} = (1+\sqrt{2})^n \ \ \ \mbox{ for } \ n= 1, 2, 3, \ldots. $$ Then how to show that (a) $x_{n+1} = y_{n+1} + y_n$, $\ \ \ ...
7
votes
2answers
82 views

How would you explain a quadratic field to a beginner?

How would you explain a quadratic field to a beginner? Eg. how did the subject first start? All the modern stuff they use to explain it makes it really confusing how one should think about it in more ...
2
votes
2answers
45 views

How to find all integer solutions for underdetermined sytsem of linear equations

I do have a system of n equations with m variables where m > n with integer coefficients. I wish to find a set of integer solutions to this system (In my case n = 2 and m = 4). Could somebody tell me ...
3
votes
3answers
50 views

How many subsets of A={1,2,3,…,10} have the property that the sum of their elements is $\geq 28$?

I've already known that the desired answer is 512. But, how can I get this answer? Can anybody show me how to get this answer with only using permutation or combination? I can only think that the ...
0
votes
2answers
30 views

Not able to understand the procedure used to find GCD of two numbers through Euclid's algorithm.

Ok so I was just touring through the basic concepts of number theory and then this doubt suddenly hit me. We use Euclid's algorithm to find the GCD of two numbers, $a$ and $b$. First step: $a=b\times ...
3
votes
1answer
34 views

Number of Divisors of most numbers

In the book A Comprehensive Course in Number Theory by Alan Baker. The author mentions that even though the average order of $\tau(n)$ is $\log n$, almost all numbers have about $(\log n)^{\log 2}$ ...
1
vote
0answers
56 views

Minimizing over partitions $f(\lambda) = \sum \limits_{i = 1}^N |\lambda_i|^4/(\sum \limits_{i = 1}^N |\lambda_i|^2)^2$

I'm trying to characterize the behavior of the the quantity: $$A = \frac{\sum \limits_{i = 1}^N x_i^4}{(\sum \limits_{i = 1}^N x_i^2)^2},$$ subject to the constraints that $$ \sum \limits_{i = 1}^N ...
2
votes
2answers
54 views

Degree $2$ nilpotent matrices with non-zero product

Let $n$ be sufficiently large positive integer. Let $M_i$ be $n$ matrices with entries in either $\mathbb{C}$ or $\mathbb{Z}/n \mathbb{Z}$. Is it possible, (1) for $1 \le i \le n, M_i^2=0$ and ...
3
votes
0answers
36 views

Are there any known special properties of a number located between twin primes?

With the exception of $4$, every number located between twin primes is divisible by $6$. This one is obvious, but are there any other properties that can be ascribed to such numbers? A property may ...
2
votes
1answer
23 views

How do we determine the decomposition of $p\mathcal{O}_K$ in $K = \mathbb{Q}(\sqrt[3]{5})$?

Let $K = \mathbb{Q}(\sqrt[3]{5})$, and $\mathcal{O}_K$ be its ring of integers. In general, how do we decide the decomposition of $p\mathcal{O}_K$, for an odd prime $p$? I know that by Kummer's ...
2
votes
1answer
52 views

Using Fermats prime numbers to prove that there is infinitely many prime numbers

A Fermat number $F_n$ is of the form $F_n = 2^{2^n} + 1$ Furthermore, $F_n = 2 + F_0F_1F_2......F_{n-1}$ Now I already proved that if $n \neq m$ then $\gcd(F_n,F_m) = 1$ Here is the proof Without ...
1
vote
4answers
60 views

What is the remainder if you divide $2^{804}$ by 257

What is the remainder if you divide $2^{804}$ by 257. I got an answer 16. Would that be right? I split the denominator 257 to be $(2^8+1)$. I split the numerator $2^{12}$ times ...
1
vote
1answer
32 views

$\bigg(\frac{-2}{p}\bigg)= \begin{cases} 1 & \text{ if $p\equiv 1$ or $3 \mod 8$} \\ -1 & \text{ if $p\equiv 5$ or $7 \mod 8$} \\\end{cases}$

Show that $$\bigg(\frac{-2}{p}\bigg)= \begin{cases} 1 & \text{ if $p\equiv 1$ or $3 \mod 8$} \\ -1 & \text{ if $p\equiv 5$ or $7 \mod 8$} \\\end{cases}$$ $\textbf{Proof:}$ \begin{equation*} ...
6
votes
3answers
197 views

I finally understand simple congruences. Now how to solve a quadratic congruence?

Now that I have plain old congruences, $19x\equiv 4 \pmod {141}$ for example, I am trying to wrap my brain around quadratic ones. My textbook shows how to tackle the aforementioned congruences, but ...
0
votes
0answers
17 views

Condition under which Hurwitz quaternion has left or right gcd equal to 1 with its conjugate

Correct me if I am wrong but for Gaussian integer - $a +bi$ its $gcd (a+bi, a-bi) = 1$ - when $gcd (a,b)=1$ and $a +bi \neq 1+i$ I wonder if there are any known conditions under which Hurwitz ...
0
votes
0answers
24 views

number theory problem. I think it should solve by Quadratic Residue. Please help me with this. [on hold]

For which primes p is −2 a square mod p. Give the answer in terms of specific congruences p must satsify.
1
vote
3answers
42 views

gcd of an infinite subset of naturals

I don't know if it's true, I would like to prove (or disprove) if $S\subset \mathbb N$ is an infinite set, then there exists an finite set $S'\subset S$, such that $\text{gcd}\ S'=\text{gcd}\ S$. I ...
0
votes
2answers
21 views

Extension of Completions of Number Fields

On p. 116 of Milne's notes on Algebraic Number Theory, he gives the following construction. Let $K$ be a field with a valuation $|\cdot|$ (archimedean or discrete nonarchimedean), and let $L$ be a ...
1
vote
0answers
24 views

The exponent on Thue's theorem

I have been reading about Runge's theorem on diophantine approximation Theorem. Let $\xi$ be an algebraic real number of degree $d\geq 3$. For every $\epsilon >0$ there is a number $\gamma >0$ ...