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.

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4
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0answers
46 views

Finding first n so nth fibonacci is c modulo p

This is a question I stumbled upon in an online programming contest archive. The problem statement is simple, given $c \equiv F(n)$ mod $P$ and $P$, where $P$ is a prime of form 5$k$ + 1 or 5$k$ - 1, ...
3
votes
2answers
30 views

Show $ \frac{1}{n} \sum_{p \in \mathbb P} \left\lfloor \frac{n}{p} \right\rfloor \log p = \log n + O(1)$

Any hints how to prove for $n \in \mathbb N$ $$ \frac{1}{n} \sum_{p \in \mathbb P} \left\lfloor \frac{n}{p} \right\rfloor \log p = \log n + O(1) $$ where $\mathbb P$ denotes the set of all primes? As ...
2
votes
3answers
50 views

the product of an odd perfect number and some even perfect number is perfect

If $a$ were an odd perfect number ,does there exist an even perfect number $b$ such that $ab$ is a perfect number?
3
votes
1answer
44 views

Can $\sigma(n)-n$ be a proper divisor of $n$?

Let $n\ge 2$ be a natural number, $\sigma(n)$ the sum of its divisors. Can $\sigma(n)-n$ be a PROPER divisor of $n$ ? If $\sigma(n)-n=n$ , $n$ is a perfect number. If $\sigma(n)-n=1$ , $n$ is a ...
2
votes
3answers
57 views

Understanding $p$-adic fields

OK, I'm completely lost on this. Define the $p$-adic integers $\mathbb{Z}_p$ as the projective limit $$\lim_{\leftarrow} \mathbb{Z}/p^n \mathbb{Z}.$$ So, if $a \in \mathbb{Z}_p$, then $a$ can be ...
1
vote
0answers
15 views

Generalized Bezout's Identity, subjected to a constraint on coefficients

I would like to know if there exists any good methods that can determine the following class of problems: Suppose there exists $n$ given positive integers $y_1, y_2, \dots, y_n$ and positive integers ...
5
votes
1answer
36 views

Is every non-archimedean absolute value on a number field equivalent to a $|\cdot|_{\mathfrak{p}}$?

Let $K$ be an algebraic number field, i.e. a finite field extension of $\Bbb{Q}$. I would like to prove that every non-archimedean absolute value on $K$ is equivalent to $$ |x|_{\mathfrak{p}} := ...
2
votes
1answer
24 views

Analytic geometry line segments

This is a very interesting analytic geometry math problem that I came across in an old textbook of mine. It is quite nice and I decided I would share it with MSE for future reference and a fun time?! ...
3
votes
0answers
61 views

'Randomness' of inverses of $(\mathbb{Z}/p \mathbb{Z})^\times$

Suppose you are given the group $(\mathbb{Z} / p \mathbb{Z})^{\times}$, where $p$ is prime. Let $A_p$ denote the sequence whose $j$th element is the inverse of $[j]$. For instance, if $p = 7$, the ...
0
votes
0answers
57 views

Some intuition about modular multiplicative inverses

I'm trying to gain some intuition about modular multiplicative inverses. Suppose I have a prime number $n$ and the group $(\mathbb{Z} / n \mathbb{Z})^\times$, and I pick some constant $\alpha$ much ...
0
votes
0answers
16 views

Multiplicative property for the coefficients of cusp form

This question might seen as a duplicate of this, however my aim is to understand the theory which lies beneath the computations of Sage. Let $\Gamma_0(4)$ be a congruence subgroup of ...
3
votes
1answer
36 views

Ratio of fruits

This is a very interesting Diophantine equation word problem that I came across in an old textbook of mine. It is quite nice and I decided I would share it with MSE for future reference and a fun ...
0
votes
3answers
32 views

Palindromic numbers, and $3$. [closed]

Let $x \in \mathbb{Z}$, and $x'$ be the palindrome of $x$. Why is it that $3 \mid (x - x')?$
1
vote
0answers
79 views

$p$ be a prime number , then is it true that $p^2| {pa \choose pb}-{a \choose b} , \forall a,b \in \mathbb N$ ?

Let $p$ be a prime number , then is it true that $p^2| {pa \choose pb}-{a \choose b} , \forall a,b \in \mathbb N$ ?
0
votes
0answers
53 views

How to find the number of subsets of a set $X$ such as the sum of their elements is divisible by 3?

Let $X$ be a set with $N$ numbers ($N$ is less than 1000). The problem is to find the number of subsets of $X$ such that the sum of their elements is divisible by 3. Lets denote this number by ...
9
votes
1answer
115 views

Integer solutions to the equation $a_1^2+\cdots +a_n^2=a_1\cdots a_n$

What is the general solution to the equation $$\sum_{j=1}^n a_j^2=\prod_{j=1}^n a_j,$$ $n\in \mathbb N$ , $n \ge 2$ over $\mathbb N_0$ ? WLOG, we can assume $0\le a_1 \le a_2\le \cdots \le a_n$ For ...
2
votes
0answers
79 views

Project Euler Problem #87 - Prime power triples

I found this problem in Project Euler: https://projecteuler.net/problem=87 The smallest number expressible as the sum of a prime square, prime cube, and prime fourth power is $28$. In fact, ...
1
vote
1answer
18 views

Does the field norm commute with field morphisms?

Consider a field extension $K$ of $\Bbb{Q}$, a finite extension $L$ of $\Bbb{Q}_p$ for some fixed prime number $p$, and a field morphism $\sigma \colon K \to L$ such that the diagram $\require{AMScd}$ ...
1
vote
1answer
20 views

$\Gamma_0(4)$ has no torsion except $-1$

Let $\Gamma_0(4)$ be a congruence subgroup of $SL(2,\mathbb{Z})$ defined as $$\Gamma_0(4)=\{M=\begin{pmatrix} a &b\\ c& d \end{pmatrix}\in SL(2,\mathbb{Z})\mid c=0\bmod 4\}.$$ How to prove ...
4
votes
1answer
30 views

$L(1,\chi) = \sum_{n=1}^{\infty}\frac{\chi (n)}{n} > 0$, for $\chi$ be the non-trivial real character

Let q be an odd prime and $\chi$ be the non-trivial real character modulo q. I am trying to prove that $L(1,\chi) = \sum_{n=1}^{\infty}\frac{\chi (n)}{n} > 0$. Note: this question was first asked ...
0
votes
2answers
40 views

Proving if $-1 < x < 1$ then $x^1 + x^2 + \cdots + x^n = \frac{x-x^{n+1}}{1 - x}$

Let $$S_n = x + x^2 + x^3 + \cdots + x^n$$ then $$xs_n = x^2 + x^3 + \cdots + x^n + x^{n+1}$$ This is taken from book "An concise introduction to pure mathematics" : Why does inserting $x$ to ...
0
votes
0answers
43 views

a question designed in number theory

sorry my friends I designed a question that I know the answer of but I don't know if it is a good question or not. Maybe the answer of this question is trivial. If it is possible give me your ...
3
votes
1answer
38 views

Elliptic curves, reduction map, $E_n$

Let $E$ be the elliptic curve and set $\phi: E(\mathbb{Q}_p) \rightarrow E(\mathbb{F}_p)$ to be the reduction morphism. Define $E_n := \{(x:y:z) \in \ker \phi | x/y \in p^n\mathbb{Z}_p\}$. I'm busy ...
1
vote
1answer
23 views

Clarification of a proof of Eisenstein's lemma

I'm working on a proof of quadratic reciprocity following Wikipedia's proof via Eisenstein, and one line in the proof seems unjustified: On the other hand, by the definition of $r(u)$ and the ...
2
votes
1answer
37 views

Is the Champernowne constant an automatic number?

The Champernowne constant in base $b \geq 2$ is obtained by concatenating the $b$-ary expansion of every integer. For example, in base $10$ this is $$ 0.123456789101112131415\dotsc $$ Question: Is the ...
0
votes
1answer
42 views

Subset Sum Problem (general)

I have a problem which turned to be similar to the Subset Sum Problem. The main differences between the Subset Sum Problem and my problem are: 1- The vector elements can be positive and negative (in ...
3
votes
0answers
18 views

NP-hardness of solving congruence equations in several variables

You are given the following equation modulo $N$ (where the $\beta_i$'s are given integers modulo $N$, and the $x_i$'s are unknown integers modulo $N$): $$\beta_1x_1 = \beta_2 x_2 = \ldots = \beta_l ...
3
votes
2answers
98 views

Why is $\sigma_1(0)$ not $-\frac{1}{12}$?

The Eisenstein series $\mathbb{G}_2$ is given by $$\mathbb{G}_2(z) = -\frac{1}{24} + \sum_{n=1}^\infty \sigma_1(n) q^n$$ with $q=e^{2\pi i z}$ and $$\sigma_1(n):=\sum_{d\mid n} d$$ for $n\in\mathbb ...
1
vote
1answer
42 views

Average patients waiting time in dental office

At a Dental Office, patients come in at the rate of 20 per hour and, on average, are processed at the same rate. Patients wait in a queue till they are called up to the counter for registration. After ...
2
votes
1answer
40 views

Coefficient in the Fourier expansion of the cusp form

Ideal of cusp for $\Gamma_{0}(4)$ is principal and generated by $f(z)=η(2z)^{12}=q+\sum a(n)q^n $, this is discussed here. How one can compute the coefficient $a(n)$ when $n$ is rather large ? for ...
1
vote
0answers
38 views

How to solve the equation $x^2+Dy^2=\alpha$ over finite fields

It is known that the equation $x^2+Dy^2=1$ is solved over finite fields $\mathbb{F}_q$ and we can point out the solutions . I wonder can we give solutions for the equation $x^2+Dy^2=\alpha$ for any ...
0
votes
3answers
75 views

Proving “The sum of n consecutive cubes is equal to the square of the sum of the first n numbers.”

From http://www.themathpage.com/aPreCalc/mathematical-induction.htm states : should : not be : $$1^3 + 2^3 + 3^3 + ... + n^3 = \frac{n^3+(n + 1)^3}{2^3}$$ as everthing to left of equation is ...
3
votes
2answers
421 views

Is it a composite number? [duplicate]

How do I prove $19\cdot8^n+17$ is a composite number? Or is that number just a prime? So I tried to find a divisor in the cases $ n = 2k $ and $ n = 2k + 1 $. But I had no success. Do you have any ...
-3
votes
2answers
71 views

What is the largest 2-digit prime factor of the integer $\binom{200}{100}$ [closed]

What is the largest 2-digit prime factor of the integer $\dbinom{200}{100}$?
2
votes
5answers
110 views

Prove that if $n^2$ is odd then $n$ is odd?

Here is my solution: I assume $n^2$ is odd then I put $n^2$= $(2x-1)^2$, now I am taking root square for both sides: $\sqrt{(n^2)}$ = $\sqrt{(2x-1)^2}$ $\Rightarrow$ $n = (2x-1)$ $\Rightarrow$ $n$ is ...
1
vote
0answers
42 views

Is it true that $(1+\varepsilon)\pi(x+y)\ge\pi(x)+\pi(y)$?

Recently I was going through Udrescu's result concerning the inequality $\pi(x)+\pi(y)\ge\pi(x+y)$ which states that ($\pi(x)$ denotes the prime counting function), For all $\varepsilon>0$ and ...
0
votes
1answer
22 views

Proving that a binomial coefficient involving a power of $2$ is even

In the process of proving that the polynomial $x^{2^n} + 1$ is irreducible in $\mathbb{Z}[x]$, I am getting stuck on proving an intermediate result: Denote $f(X)=X^{2^n}+1.$ By a linear change of ...
0
votes
3answers
80 views

How to find number which is greater?

I have two numbers $\left(\frac{8}{7}\right)^{\frac{7}{8}}$ & $\left(\frac{9}{8}\right)^{\frac{8}{9}}$. How to find out the greater number? I am 12th grad. thanks for your help.
4
votes
2answers
56 views

Help with proving that the torsion subgroup of $y^2=x^3+x$ is $E(\mathbb{Q})_{tors} \cong \mathbb{Z}/2\mathbb{Z}$

Let $E: y^2= x^3 + x$ be an elliptic curve over $\mathbb{Q}$. I'm trying to prove that $E(\mathbb{Q})_{tors} \cong \mathbb{Z}/2\mathbb{Z}$. In order to do that, I've already shown that ...
0
votes
0answers
19 views

References for Dirichlet characters and L-functions

I am working on some exercises from my Analytic Number Theory course regarding Dirichlet characters, and I was wondering if someone could provide some references for this. Here's a problem that I'm ...
4
votes
2answers
38 views

Primitive $p^n$-th root of unity in $\bar{\mathbb{Q}}_p$.

I am trying to solve the following exercise in Koblitz's "$p$-adic Numbers, $p$-adic analysis, and Zeta-Functions". Let $p$ be a prime. Let $a$ be a primitive $p^n$-th root of unity in ...
5
votes
1answer
30 views

Prove that given graph consisting of vertices numbered with composite numbers is not eulerian

We have the following graph definition: $$V(G_n)=\{1\leq m\leq n : m = pq\}$$ (so vetices of $G_n$ are composite numbers) $$E(G_n)=\{\{i,j\}:i\perp j\}$$ (so vertices $i,j$ are connected if and only ...
1
vote
3answers
51 views

Solutions of $y^2 = \alpha$ in $\mathbb{F}_{19}$

So I'm working on an exercise for elliptic curves and in one of my steps I have to determine all numbers $y \in \mathbb{F}_{19}$ for which it holds that $y^2 = \alpha$, with $\alpha \in ...
7
votes
1answer
82 views

On the solvability of the negative Pell equation $x^2-2py^2 = -1$

Given prime $p=8n+1$. Then $$x^2-2py^2 = -1\tag1$$ is not solvable for, $$p_1= 17, 73, 89, 97, 193, 233, 241, 257, 281, 337, 353, 401, 433, 449, 577, 593,601, 617, 641,\dots$$ but is solvable ...
2
votes
0answers
70 views

Does the Euler product for the Dirichlet $\beta$-function converge for all $\Re(s)>\frac12$?

The Dirichlet $\beta$-function is defined for $\Re(s)>0$ as: $$\beta(s) = \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^s}$$ It has the following Euler product (I used that Dirichlet character ...
2
votes
0answers
19 views

What is binary norm of quadratic fields of sum of two squares such that one of them is necessarily even like $a^2 +4b^2?$

I am trying to simplify an expression which I have reached, suppose a number can be represented in the form of $D=a^2 + 4b^2$. What is binary norm of $D$, or how else can it be represented?
6
votes
1answer
35 views

Where $ax + b$ prime infinitely often, is $ax + b - 2$ semiprime at least once?

I'm trying to figure out a way to prove this: Given arithmetic progression $ax + b$ where $a$, $b$ coprime and $ax + b$ is prime infinitely often, it is the case at least once that $ax + b - 2$ is ...
6
votes
1answer
63 views

Is there any general method for solving $(a_1+a_2+..a_n)^2=a_1^3+a_2^3+…+a_n^3$ in positive integers $a_1,a_2,…a_n$?

We know the identity $(1+2+...+n)^2=1^3+2^3+...+n^3$ . So I was thinking , for given $n\in \mathbb N$ , is there any general method for solving $(a_1+a_2+..a_n)^2=a_1^3+a_2^3+...+a_n^3$ in positive ...
-1
votes
1answer
36 views

A question on polynomial congruence

If $m\ne 4$ is a composite number, then does there exist two integers $a$ and $b$ which, divided by $m$, give a remainder different from zero and such that if $f(x)$ is a polynomial with integral ...
1
vote
1answer
51 views

$P(n)$ is the product of two digits in the integer $n$

$P(n)$ is the product of two digits in the integer $n$. For example, $P(18)=1\cdot8=8$ $P(50)=5\cdot0=0$ $P(99)=9\cdot9=81$. Is there any nice way to find the value of ...