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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3
votes
2answers
213 views

a general continued fraction satisfying $\frac{(i+\Theta\sqrt{z})^m}{(i-\Theta\sqrt{z})^m}=\frac{(ik+\sqrt{z})^{m+1}}{(ik-\sqrt{z})^{m+1}}$

Given any natural number $m\gt2$, let $z$,$k$ be complex numbers, where $i=\sqrt{-1}$ and consider the general continued fraction $$\Theta(k,z,m)=\cfrac{(m+1)}{km+\cfrac{z(0m-1)(2m+1)} ...
1
vote
1answer
56 views

hailstone sequence of perfect squares (collatz conjecture)

The Collatz conjecture states: Take any positive integer $n$. If $n$ is even, divide it by $2$ to get $n/2$. If $n$ is odd, multiply it by $3$ and add $1$ to obtain $3n + 1$. Repeat the process ...
1
vote
1answer
48 views

Co prime numbers

How to find a set of numbers which are coprime to each other (all numbers are pairwise co prime) ? the numbers can be assumed to be less than a specific integers. like all numbers in the range [0,x]
1
vote
3answers
80 views

Congruence $16^{(x^ 2+x+1)} \equiv 4 \mod 11$

Given the congruence $16^{x^2+x+1}≡ 4 \mod 11$ I'm not necessarily sure how to approach this problem if someone can help me head in the right direction since 16 is not a primitive root of mod 11 I ...
1
vote
0answers
15 views

Finding a zero of homogeneous parts of polynomials using zero of those polynomials

Let $f, g \in \mathbb{Q}[x_1, ..., x_n]$ be polynomials of degree $d$. Let $F$ and $G$ denote the degree $d$ portions of $f$ and $g$ respectively. Suppose there exists a non-singular point ...
1
vote
0answers
27 views

which values of k satisfies special property to formulate L function

Consider $x*\prod_{a=1}^{n}(1-x^a)^k$ Famously for k=24 this product satisfies the condition to be an L-Function. More information can be found here My question is for what other values of k, such ...
-2
votes
1answer
24 views

Effective upper bound for a sum over prime numbers

Fix $y$ a positive real number. Is there an effective bound for the following sum i.e a positive constant B such that $$\sum_{p>y}\sum_{\nu \geq 4} \frac{1}{p^{9\nu/32}} \leq B.$$ Many thanks.
0
votes
0answers
14 views

Products of bounded numbers whose residues agree.

Given small $\epsilon>0$, $0<q<r<r+s=1$ and large enough $n\in\Bbb N$ is it always possible to find coprime $a,b$ and coprime $c,d$ such that $$a\approx n^{q+\epsilon},b\approx ...
0
votes
0answers
31 views

Find a polynomial $f(x)$ s.t. $f(x) ≡ 0 \pmod{p^e}$ has only one solution but $f(x)≡ 0 \pmod{p^{(e+1)}}$ has $p$ solutions

Find a polynomial $f(x)$ s.t. $f(x) ≡ 0 \pmod{p^e}$ has only one solution in $Z(p^e)$ but $f(x)≡ 0 \pmod{p^{(e+1)}}$ has $p$ solutions in $Z(p^e+1))$ for any $e≥1$ and odd primes. I didn't find such ...
0
votes
1answer
38 views

Base 8 to X(Base 16) conversion

475.641(Base 8) to X(Base 16) Answer is 13D.D08(Base 16) My attempt: 27BA1(Base 16) Which step I had missed? Thanks.
1
vote
1answer
43 views

Galois group action on etale cohomology groups

Let $X$ be a smooth and proper scheme over $Spec(\mathbb{Z}_p)$. Let $l$ be a prime number coprime to $p$. Then the proper base change theorem gives me an isomorphism ...
0
votes
0answers
55 views

count arrays with pairwise gcd as some integer (was a previous question by other user but no answer yet) [closed]

Array must contain N elements. Each element in array must be less than given number M. Greatest common divisor of all its elements is equal to a given integer D.
0
votes
1answer
25 views

divisibility test by induction

Some help to finish off this part.. Prove that $n(n^2 + 5)$ is divisible by $6$ for positive integer $n$. I need to show that it is also true for $n = k + 1$ . I have started well though.
2
votes
0answers
25 views

Dirichlet character of order $4$ and the splitting of $p$ in $\mathbb{Z}(\sqrt{-1})$

For $p \equiv 1 \pmod{4}$, let $\psi$ be one of the two Dirichlet characters of order $4$ in $(\mathbb{Z} / p \mathbb{Z}) ^\times$. Consider the character sum $S = \sum_{x=0}^{p-1} \psi(x^2 - a)$, ...
0
votes
1answer
49 views

If n is a positive integer that is not square free

Determine if the statement is, in general, true or false. Recall that a universal statement is true if it is true for all possible cases while it is false if there is even one counterexample. Be ...
0
votes
0answers
88 views

Analytic Bound on The Riemann Zeta Function

Given the canonical infinite product representation (Weierstrass form) of the gamma function, $$\Gamma(z)= \left [ze^{\gamma z}\prod_{m=1}^{\infty} \left ( 1+ \frac{z}{m} \right)e^{-z/m} \right ]^{-1} ...
2
votes
0answers
67 views

$m^2+n^2$ and $m^2-n^2$ cannot both be squares [duplicate]

I need to show that there aren't any $m$ and $n$ such that $m^2+n^2$ and $m^2-n^2$ are both squares. First of all, assume without loss of generality that $m$ and $n$ are co-prime, since otherwise we ...
0
votes
1answer
49 views

Proving that $n$ is a Carmichael number

Determine if the statement is, in general, true or false. Recall that a universal statement is true if it is true for all possible cases while it is false if there is even one counterexample. Be ...
0
votes
1answer
25 views

lower bound for Chebyshev first function [closed]

Is there a method to provide the a lower bound for $$\vartheta(x)=\sum_{p\leq x}{\ln p},$$ where $p$ runs over the primes ? Also, does anyone know of citations for the famous Chebyshev first ...
2
votes
1answer
21 views

when p is sum of two square integer, prove (a/p) which is legendre symbol = 1

Let a,b be integers and p be an odd prime. if $p$=$a^2+b^2$ and a is odd, prove $(a/p)$ which is legendre symbol = $1$ what i have done is that : because p and a are odd, b must be even and p is ...
2
votes
0answers
42 views

Write as a product of integers [closed]

My question is that in how many ways can $10,000!$ be written as the product of $30$ distinct positive integers. My question is similar to this question: In how many ways can $1000000$ be expressed as ...
1
vote
0answers
19 views

Pattern in Repeated Digits of Square Numbers

Is there a significance to the length of the repeating pattern of the ones, tens, hundreds... place of consecutively ascending square numbers?
6
votes
2answers
109 views

Prove that $2^x = 3 \cdot 9^m+5$ has no positive integer solutions for $m \geq 2$

Prove that $2^x = 3 \cdot 9^m+5$ has no positive integer solutions for $m \geq 2$. I noticed that $x \equiv 5 \bmod 6$ and thus $2^x \equiv 4 \bmod 7$, but that doesn't seem to help me since $3 ...
0
votes
1answer
17 views

Finding possible solutions to a modular arithmetic question.

How many solutions does the quadratic equation $\bar{x^{2}}= \bar{2}$ have in $\mathbb{z}_{47}$? The only way i can think of doing this is computing all the squares within the modulo. But i know ...
2
votes
2answers
115 views

When is $\frac{a}{n} = n^a$ true?

Title says it all really, I am trying to figure out if theres a situation where $\frac{a}{n} = n^a$ is true or if this is impossible. This is not realy from somewhere, just for the sake of curiosity. ...
2
votes
3answers
85 views

Show that $\forall n\in\mathbb{N}$, $14^n$ can be represented as a sum of three perfect squares.

Show that $\forall n\in\mathbb{N}$, $14^n$ can be represented as a sum of three perfect squares. I checked $(\mod 7)$ and deduced that the three squares can be $1,4,2(\mod 7) $ or all divisible by ...
0
votes
1answer
14 views

Solve congruence equation using primitive roots and indices

I am having trouble solving the equation $ x^7 \equiv 3 \; \pmod 9 $ using primitive roots and theory of indices. Can someone please help? I got $ 2 $ and $ 5 $ are primitive roots of $ 9 $ by ...
-2
votes
2answers
140 views

What numbers are anti-divisors of $105$?

I don't get the mean of anti-divisor right. By this page, the number of anti-divisors of $105$ is $9$. However, what numbers are anti-divisors of $105$? Would you teach me the process of the ...
1
vote
3answers
129 views

Find all integral solutions of the equation $x^n+y^n+z^n=2016$

Find all integral solutions of equation $$x^n+y^n+z^n=2016,$$ where $x,y,z,n -$ integers and $n\ge 2$ My work so far: 1) $n=2$ $$x^2+y^2+z^2=2016$$ I used wolframalpha n=2 and I received ...
5
votes
1answer
77 views

Equality of power towers : $a\uparrow\uparrow m=b\uparrow \uparrow n$

Suppose, $a,m,b,n$ are natural numbers greater than $1$. If we have $$a\uparrow\uparrow m=b\uparrow\uparrow n$$ can we conclude $a=b$ and $m=n$ ? $a\uparrow \uparrow m$ is a powertower of $m$ $a's$ ...
1
vote
1answer
19 views

Modular zero of a function

Is there a quick and dirty way to find the modular zero of a function, such that f(x) = 0 (mod p), p being a prime. E.g. f(x) = x^2 - 5x + 1 and p = 7. It's quite easy to find that for x = 6 the ...
0
votes
0answers
20 views

Relation between the eigenvalues of $\Delta$ and counting lattice points

I was reading a paper with the following information: "Let $\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n$ be the flat torus, let $\varphi$ be the eigenfunctions and $\lambda$ the eigenvalues of the ...
3
votes
2answers
47 views

Show that $a^{13} \equiv a \pmod{3 \cdot 7 \cdot 13}$.

Show that $a^{13} \equiv a \pmod{3 \cdot 7 \cdot 13}$. I want to know if my attempt is correct. First $a^{13} \equiv (a^3)^4 \cdot a \equiv a^4 \cdot a \equiv a^3 \cdot a^2 \equiv a \cdot a^2 ...
-1
votes
1answer
44 views

sum of squares equals the product of their roots [closed]

I was doing some math about a three by three magic square of squares, and I managed to prove that there are some numbers $\alpha$ and $\beta$ such that $$\alpha^2 + \beta^2=\alpha\beta$$ Is this ever ...
1
vote
0answers
27 views

Are there curves of genus 2 and higher over number fields with everywhere good reduction?

a theorem of Fontaine states that there are no curves of genus $\geq 1$ over $\mathbb Q$ with everywhere good reduction. For curves of genus one over number fields, this is not true. There are number ...
4
votes
2answers
73 views

Algebraic number fields in which all rational primes are inert

Is there an algebraic number field $F\supsetneq\mathbb{Q}$ such that all rational primes are inert in $\mathcal{O}_F$?
2
votes
0answers
33 views

show that number theory inequality $a_{1}+a_{2}+\cdots+a_{k}<\frac{5}{2}n$

Let $n,a_{1},a_{2},\cdots,a_{k}$ be postive integers and at least greater than $1$,and such $$(a_{1})!\cdot(a_{2})!\cdots(a_{k})!|n!$$ show that $$a_{1}+a_{2}+\cdots+a_{k}<\dfrac{5}{2}n$$ I have ...
0
votes
0answers
25 views

Fractions ( k / p - k ) ≠ prime/prime imply gcd(k / p - k) = 1

Prove that for every $$k ∈ \{{ 1, 2, 3, ..., \frac{(p - \frac{1 - (-1) ^ p}{2} )}{2}}\}\E $$(E is the set of even numbers) such that $$\frac{k}{p - k} \neq \frac{prime}{prime}$$ implies that for all k ...
0
votes
0answers
29 views

Find Expected Value, Variance, and Limit of Uniform Distribution

Let $X_1, X_2, \ldots, X_n$ be a sequences of independent random variables. $X_i \sim U(0, 2A)$. Compute $E(X_i)$ and the $Var(X_i)$. Also compute the $lim_{n\to\infty} P(X_1, X_2, \ldots , X_n > ...
0
votes
3answers
72 views

For what values is $x + y = z$ and $a^x + a^y = a^z$?

I came across this question while trying to solve an equation with a friend. Obviously the statement $$x + y = z \implies a^x + a^y = a^z$$ Is false. However, I'm a bit curious if there are any other ...
-1
votes
0answers
13 views

A number of 0,1-sequences

Could you find a number of binary sequences length n,which is not more than r units in a row? For example, number of these sequences A(n,r) for n=2 A(n,r)=1 if r=0; A(n,r)=3 if r=1; A(n,r)=4 if r=2;
0
votes
1answer
24 views

Using the Law Quadratic Residues

I'm looking to find if $\bigg(\frac{1210}{41}\bigg)$ so is 1210 a quadratic residue modulo 41. So I have ...
2
votes
1answer
49 views

How many real solutions are there to the equation $x = 1964\sin{x}-189$?

How many real solutions are there to the equation $x = 1964\sin{x}-189$? Attempt: Graphing $y_1=x$ and $y_2=1964\sin{x}-189$ we see that at $x = 1775$ we achieve the maximal height for $y_2$ and ...
0
votes
0answers
40 views

congruence solution

Determine is, in general, true or false. Recall that a universal statement is true if it is true for all possible cases while it is false if there is even one counterexample. Be prepared to prove that ...
4
votes
1answer
63 views

If $a$ is not divisible by $7$, then $a^3 - 1$ or $a^3 + 1$ is divisible by $7$

Determine is, in general, true or false. Recall that a universal statement is true if it is true for all possible cases while it is false if there is even one counterexample. Be prepared to prove that ...
0
votes
0answers
62 views

If $m =4^{n +1}$ with $n>0$ and m is prime then $3^\frac{m-1}{2}$ =-1(mod m)

Determine is, in general, true or false. Recall that a universal statement is true if it is true for all possible cases while it is false if there is even one counterexample. Be prepared to prove that ...
0
votes
1answer
30 views

Let p be a odd prime, If ord p (a) = h and h is even, then a^(h/2)= -1 mod p

Determine is, in general, true or false. Recall that a universal statement is true if it is true for all possible cases while it is false if there is even one counterexample. Be prepared to prove that ...
0
votes
2answers
39 views

Find a Coefficient of an expansion

I have given expression $(X+X^a+X^b+X^c+X^d+....)^k$ where $a$, $b$, $c$ ... and $k$ are whole numbers. How we can find the $Z$ coefficient, i.e, $X^Z$ of the above term $1\le Z \le k$? For Example ...
2
votes
2answers
43 views

What function when given the inputs $x,y$ returns the given $z$? [closed]

What function when given the inputs $x, y$ returns the given $z$? When $x = 2, y = 10$, $z = 1$ When $x = 6, y = 10$, $z = 2$ When $x = 50, y = 70$, $z = 5$ When $x = 16, y = 17$, $z = 1$ When $x = ...
3
votes
4answers
103 views

Prove that $\displaystyle \sum_{k=1}^n \bigg(\dfrac{1}{k}+\dfrac{2}{k+n}\bigg ) \leq \ln(2n) + 2 -\ln(2)$

Prove that $$\displaystyle \sum_{k=1}^n \bigg(\dfrac{1}{k}+\dfrac{2}{k+n}\bigg ) \leq \ln(2n) + 2 -\ln(2).$$ I was thinking of using mathematical induction for this. That is, We prove by ...