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)

0
votes
0answers
7 views

Integers points of an elliptic curve

I am concerned by the number N of integer points in some class of elliptic curves. It is known to be finite for each curve C the corresponding bound being a function $N_C$ which gives a huge number. ...
2
votes
0answers
6 views

number of weak compositions modulo prime $p$

For $n\in \mathbb{N}$ and some prime $p$, consider $(\mathbb{F}_p)^n$. Is it known how many weak compositions $$x_1+x_2+\ldots +x_n\equiv 0 \pmod p$$ in $\mathbb{F}_p$ there are, where $(x_1, \ldots, ...
1
vote
0answers
5 views

Depict Complex Plane of Quadratic Field

Can someone show me a complex plane around the origin, with the points on the part of the complex plane which are quadratic integers in $Q[√−1]$. Another graph for $Q[√−3]$. And another for $Q[√−5]$. ...
1
vote
1answer
24 views

Is there a fast divisibility check for a fixed divisor?

Is there a fast algorithm to check if $d \mid n$ is true for varying $n$, if divisor $d$ is fixed? Variable $n$ is a $w$-bit binary integer, $d$ is an integer constant.
1
vote
1answer
26 views

Prove that $\phi(n)=\frac{n}{2}$ iff $n=2^k$ for some integer $n$

Prove that $\phi(n)=\frac{n}{2}$ iff $n=2^k$ for some integer $n$ Attempt: Let $n=p_1^{\alpha_1}p_2^{\alpha_2}\dots p_k^{\alpha_k}$. Then $\phi(n)=\frac{n}{2} \implies ...
1
vote
1answer
9 views

What other types of distributivity are there?

When I say ‘Distributivity,’ I mean the way a number $x$ can be ‘Put in to’ some other function or the like. For example, to distribute $x$ into $\text{id}_y$, you simply have ...
-1
votes
2answers
42 views

A bizarre property

Studing some fact about p-adic numbers I read a bizarre property. A metric space S is called ultrametric when $d(x, y) \le\max\{d(x, z), d(z, y)\} \forall(x,y,z) \in S^3$. Prove that all ball of S ...
0
votes
1answer
12 views

Primes in Quadratic Fields with Norm less than 6

What are the primes in $\mathbb Q[\sqrt{−1}]$ which have norm less than $6$? Also what primes in $\mathbb Q[\sqrt{−3}]$ have norm less than $6$, and the primes in $\mathbb Q[\sqrt{−5}]$? Which of them ...
0
votes
2answers
35 views

What do you call the subset of all the Gaussian integers having the smallest magnitude given an argument?

How is called the subset of Gaussian integers such that from all Gaussian integers having the same argument only one with the smallest absolute value is included? Is there a special name for them? ...
0
votes
1answer
20 views

Show that $ζ$ is a Quadratic Integer in $Q[\sqrt{−3}]$

So in the complex plane, there are three cube roots of one. Suppose we let $ζ$ be the cube root of one which has positive imaginary part. How can we show that $ζ$ is a quadratic integer in ...
0
votes
1answer
37 views

How Deficient a Number is? (Finding numbers having a certain deficiency)

This question was edited, in particular equations were corrected: A number N is said to be deficient by an integer $d$ if: $\sigma(N)=2N-d$ Note that powers of 2 are deficient by 1. While a prime ...
0
votes
2answers
29 views

Purely number theory problems

Suppose the numbers $1,2,3,\dots,1986$ in any order are concatenated then prove that the number is not a perfect cube. This problem gives me a feeling that here cubic residues can only help no other ...
2
votes
1answer
25 views

Average difference between two odd numbers of equal length

If I select two different odd numbers of binary length $l$, what is the formula that will tell me the average difference between those two numbers? Note that the high order digit must always be $1$, ...
10
votes
1answer
139 views

A proof involving the Euler phi function

Problem: Let $\varphi$ be the Euler phi function, where for any $n \in \mathbb{Z^+}$, $\varphi(n)$ is the number of positive integers less than $n$ that are relatively prime with $n$. ...
0
votes
1answer
37 views

What is the number of complex integers inside a circle of radius r? [on hold]

What is the number of such complex integers, $z$, that $|z|\le r$? I am interested in a closed-form formula for integer $r$.
-1
votes
2answers
44 views

Solving $a^2$ $+$ $ b^2$ $=$ $2c^2$ [on hold]

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 ...
7
votes
4answers
161 views

Understanding the trivial primality test

I'm reading an algorithms book and I came across a code example for a primality test. The problem is that I couldn't understand the condition for the for-loop: ...
0
votes
0answers
23 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?
7
votes
3answers
92 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
28 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
53 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
65 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
63 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
47 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: ...
2
votes
0answers
10 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
23 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
48 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
73 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
15 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
27 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 ...
1
vote
2answers
54 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
54 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
83 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 ...
3
votes
3answers
51 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
51 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 ...