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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5answers
33 views

When $p=2$ or $p$ prime, with $p=1\pmod{4}$, $x^2\equiv -1\pmod{p}$ is soluble - trouble understanding proof

Theorem: When $p=2$ or $p$ prime, with $p=1\pmod{4}$, $x^2\equiv -1\pmod{p}$ is soluble Proof: When $p=2$, the statement is clear. Assume $p\equiv 1\pmod{4}$, let $r=\frac{p-1}{2}$ and $x=r!$ Then ...
7
votes
0answers
49 views

A comment in the Disquisitiones Arithmeticae

Gauss proves that if $t\equiv\pm 3\mod 8$, then $2$ is a non-(quadratic)-residue modulo $t$ as follows: Assume $t\equiv\pm 3\mod 8$ is the smallest counter-example, and say $a^2\equiv 2\mod t$, ...
0
votes
2answers
32 views

Polynomial Congruence problem

We are asked to find the solutions to the following congruence $$ x^3 + 8x^2 - x - 1 \equiv 0 \ (\text{mod } 11). $$ I know that the solution can be computed using Hensel's Lemma or by simply using ...
1
vote
1answer
14 views

What does it mean by “level sets of $\bar{G}$, a collection of forms, partition those of $\bar{F}$, another collection of forms”

I was reading an article and I was wondering if someone could explain me what a certain phrase meant. Let $\bar{F}$ be a collection of integral forms of degree less than or equal to $d$. And suppose ...
4
votes
1answer
96 views

Why is it called the Fundamental Theorem of Arithmetic?

The Fundamental Theorem of Arithmetic is easy enough to understand, saying that every integer greater than 1 is either prime or is the product of a unique combination of prime numbers. What I don't ...
4
votes
1answer
63 views

Maybe is right $\frac{n^2 + 1}{4k + 3} \notin \mathbb{Z}, n, k \in \mathbb{N}^{+}$

Prove or disprove $$\dfrac{n^2 + 1}{4k + 3} \notin \mathbb{Z}, n, k \in \mathbb{N}^{+}$$ I know if $n^2 + 1$ is prime if and only if $n^2 + 1 \equiv 1 \pmod 4$.
1
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2answers
27 views

How can a subgroup have multiple cosets?

I am currently reading An Introduction To The Theory Of Groups, by Joseph Rotman, and in a section describing cosets, there is an exercise question as follows; Let $H$ be and subgroup of $G$ having ...
1
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1answer
46 views

How to count each numeral of occurrences of digits?

I want to count each numeral(0 through 9) of occurrences of digits in the range $[1, n]$. Note that 101 has two one and one zero. For example, if $n$ equals $11$: ...
1
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3answers
86 views

Can transcendental to the power transcendental be rational?

Can a transcendental number to the power of a transcendental number be a rational number?
-2
votes
1answer
25 views

Least quadratic Non residue [closed]

What are all results known yet using without using riemann hypothesis on the bounds on Least quadratic non residue .
2
votes
0answers
28 views

Titchmarsh S function

SO it is known that Titchmarsh S function $$ S(T)= \pi^{-1} arg\quad \zeta\bigg(\frac{1}{2}+iT\bigg)$$ under the assumption of *riemann hypothesis * gives $$ S(T)=O(\frac{\log T}{\log \log T})$$ can ...
2
votes
0answers
21 views

Show that $\gcd(x_1,…,x_k,x_{k+1})=\gcd(\gcd(x_1,…,x_k),x_{k+1})$

I would really appreciate if you could check my proof. Thank you. Show that $\gcd(x_1,...,x_k,x_{k+1})=\gcd(\gcd(x_1,...,x_k),x_{k+1})$ Let $\gcd(x_1,...,x_k,x_{k+1})=a$. We first show that $a$ ...
1
vote
1answer
30 views

Group of Dirichlet Characters Modulo $q$ is Isomorphic to $(\mathbb{Z} / q\mathbb{Z})^*$

I'm currently reading a book on analytic number theory, and shortly after defining Dirichlet characters, the author stated that one can prove that for a given $q\in\mathbb{N}$, the group of Dirichlet ...
2
votes
2answers
35 views

When is Chebyshev's $\vartheta(x)>x$?

Various bounds and computations for Chebyshev's functions $$ \vartheta(x) = \sum_{p\le x} \log p, \quad \psi(x) = \sum_{p^a\le x} \log p $$ can be found in e.g. Rosser and Schoenfeld, Approximate ...
1
vote
1answer
17 views

prove a function is not one-to-one

Let us look at the field $\mathbb{F}_{p}=\{0,1,2,...,p-1\}$ for a prime number p. And let $f:\mathbb{F}_{p}\rightarrow \mathbb{F}_{p}$ be the function given by $f(n)=n^2 \space (mod \space p)$. How ...
4
votes
1answer
87 views

How do I prove that the recurrence contains no perfect square?

Given the recurrence $$a_{n+2} = 14a_{n+1} - a_n - 6$$ with $a_1=1$ and $a_2=8$, how do I prove that none of the $a_n$'s apart from $a_1$ is a perfect square. This is not a homework problem, rather ...
0
votes
0answers
28 views

Diophantine eqution with a parameter

My question is about the problem when is the number $$\frac{m^3 + n^3}{n^2+m^2+m+n+c}$$ a natural number. Here $c\in \mathbb{N}$ is a constant and $m, n \in \mathbb{N}$ are the variables. This ...
-2
votes
0answers
39 views

How to uniquely write integers in rational bases

If we wish to write 13(decimal system) on base 3 system we would write it as: (1)*3^2 +(1)*3^1 +(1)*3^0 = 111(base 3) What happens if we wish to write the same number (13) in a system that uses a ...
0
votes
1answer
10 views

Solving a congruence with an invertible piece

If I have $$a \equiv bp^k \bmod p^e$$ for $0 \leq k \leq e$ with $a,p,k,e$ known. How do I solve for $b$ given that $b$ is invertible?
0
votes
0answers
29 views

Open conjectures in number theory that is easy to do some programming for

I have a to do a project in number theory that we are assigned that we should do some programming for that is not the collatz conjecture, so any suggestion would be really great.
1
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1answer
62 views

Number of real embeddings $K\to\overline{\mathbb Q}$

How many real embeddings, $K\to\overline{\mathbb Q}$ with $K=\mathbb Q\left(\sqrt{1+\sqrt{2}}\right)$ are there ? We set $f(x)=x^4-2x^2-1$ and if $\alpha=\sqrt{1+\sqrt{2}}$ then $f(\alpha)=0$. ...
0
votes
2answers
40 views

Hensel’s Lemma Number Theory Confusion

I have been given an example, finding the solutions of the congruence $f(x) ≡ 0$ (mod $5^4$) for $f(x)=x^2+1$ This solution finds that for mod $5$ we have $x_0=2$ . So through the 'lifting' process, ...
-1
votes
3answers
77 views

Interesting question in internet [duplicate]

Is this even possible to solve? 30 is an even number. I don't think there's Answer for this .
1
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3answers
63 views

Is the following a number field?

Is the field obtained by adjoining all the cube roots of $-3$ to $\mathbb Q$ a number field ? The cube roots of $-3$ are: $-\sqrt[3]{3},\sqrt[3]{3}e^{\frac{i\pi}{3}}, ...
0
votes
0answers
26 views

Fast algorithm to invert a large sparse matrix

I am interesting in sparse matrix that defined at here. I am looking for a fast algorithm to invert the matrix (better than Gaussian Elimimation). Could you suggest to me some methods that reduce ...
2
votes
1answer
83 views

Asymptotics for square-free numbers in an arithmetic progression

Set $$Q(s,\chi)=\sum_{n=1}^{\infty}\frac{\mu(n)^2\chi(n)}{n^s},\quad (s=\sigma+i\tau),$$ where $\chi$ is a character $\mod q$, Show that $Q(s,\chi)=L(s,\chi)H(s,\chi)$ where $H(s,\chi)$ is a ...
3
votes
0answers
62 views

How prove $\sigma(4^p-1)<(2^{p+1}-1)^2$

If $p$ is an odd prime numbers, show that $$\sigma(4^p-1)<(2^{p+1}-1)^2$$ where $\sigma(n)$ stands for the sum of divisors. I thought of using the formula for $\sigma(n)$: If ...
2
votes
1answer
53 views

Elliptic Function

Let $y$ be the function defined by $$y(\theta)=2sin\frac{\theta}{2}\prod_{k=1}^{\infty}\frac{(1-e^{i\theta}q^k)(1-e^{-i\theta}q^k)}{(1-q^k)^{2}}$$ where $q = e^{2\pi i\tau}$ Show that $y$ has simple ...
2
votes
1answer
21 views

Logarithm of the n'th prime.

Let $P_n$ denote the n'th prime number. How could we conclude the following from the prime number theorem? $$ \log(P_n)=\log n + \log\log n + o(1) $$ Maybe by showing that $P_n=An\log n $ for a ...
1
vote
2answers
73 views

Comparing series

Can anyone explain why if I compare the coefficient of $x^{n}$ of the equation $$\sum_{k=0}^{\infty}a(n)x^n= \frac{1}{1-x}-\frac{x}{1-x^3}+\frac{x^2}{1-x^5}-\frac{x^3}{1-x^7}+...$$ I can get ...
1
vote
1answer
26 views

Gauss sum of character $\psi \neq 1$

I am trying to solve Let $1 \neq \psi$ be a charachter of $\mathbb{F}_p$ and define $$G(\psi) = \sum_{x\in \mathbb{F}_p} \psi(x^2) $$ Proof that $|G(\psi)|^2 = p$. What I tried so far: ...
0
votes
0answers
22 views

Squareclasses in transcendental extensions of the p-adics

Let $p$ be any prime and $k = \mathbb{Q}_p$. The structure of the square class group $k^*/k^{*2}$ is well known. It has four or eight elements depending on whether $p$ is odd or not. If we set $K = ...
-1
votes
1answer
32 views

Greatest Common Divisor Integer [duplicate]

I need help with the following problem: Let $n$ and $m$ be positive integers. Prove that $$\frac{\gcd(n,m)}{n}{n \choose m}$$ is an integer.
0
votes
2answers
44 views

Prime Factorization

Let $n\ge0$. What is the power of $2$ in the prime factorization of $(2^n)!\,$? Prove that the value is correct. So far I've conjectured the value to be $2^n - 1$. This is true for $n=0,1,2,3,4$. ...
5
votes
0answers
37 views

Generalisation of $\gcd\left(\frac{a^n-b^n}{a-b},a-b\right)=\gcd(n\gcd(a,b)^{n-1},a-b)$ to $\gcd\left(\frac{a^n-b^n}{a-b},a^m-b^m\right)$?

We have the identity $$\gcd\left(\frac{a^n-b^n}{a-b},a-b\right)=\gcd(n\gcd(a,b)^{n-1},a-b).$$ (see here) This appears to be a quite useful result with various applications. I wonder whether there is ...
1
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0answers
34 views

Generator of group, Computation of discrete logarithm

The prime number $p=67$ is given. Show that $g=2$ is a generator of the group $\mathbb{Z}_p^{\star}$. Compute the discrete logarithm of $y=3$ as for the base $g$ with Shanks-algorithm. Compute the ...
3
votes
1answer
49 views

How could I get access to more than the first 2 mln non-trivial zeros of $\zeta(s)$?

I would like to test whether or not the following product (or its complement) $$\displaystyle \displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\frac12+ (-1)^n\, \gamma_n \, i} \right)$$ converges ...
4
votes
4answers
179 views

Odd divisibility induction proof

Prove that for odd $n>3$ $$64\ | \ n^4-18n^2+17$$ I checked that for $n=5$ it works. I think I need to assume that for $2n+1$ it holds and show that $2n+3$ also holds. Any ideas?
3
votes
3answers
60 views

Prime ideals lying above in $\mathbb{Q}(\sqrt{-5})$

I'm really struggling to understand the concept of prime ideals lying above and below a given prime ideal. For example taking the extension $\mathbb{Q}(\sqrt{-5})\big/\mathbb{Q}$, how do we know $(2, ...
1
vote
2answers
27 views

Generator of group, find the inverse, solve equation

Given the prime number $p=101$ Find a generator of the group $\mathbb{Z}_p^{\star}$. How many generators of $\mathbb{Z}_p^{\star}$ are there? Find $5$ generators. Find the inverse of $\overline{83}$ ...
8
votes
1answer
117 views

square numbers whose only digits are 0 and 1

Does there exist any perfect square integer (other than $10^{2k}$) whose digits are only $0$ and $1$ in base 10 expression? This just comes up in a leisure talk with my friends. Is that elementary or ...
1
vote
3answers
70 views

Find the last three digits of $17^{256}$ [duplicate]

Find the last three digits of $ 17 ^{256} $ We have to check mod $1000$ I tried to check some patterns but in vain.!
1
vote
0answers
24 views

$\max{a_1}$ in $(x_n)_{n\in\mathbb{N}}$

Let be $x_1,x_2,x_3,\ldots,$ a sequence of positive integers. Suposse the folowing conditions are true for all $n\in\mathbb{N}$ $n|x_n$ $|x_n-x_{n+1}|\leq 4$ Find the maximun value of $x_1$ I ...
0
votes
0answers
15 views

Lattices, compact orbits, and admissible boxes

A box in $\mathbb{R}^n$ is a set of the form $[-b_1,b_1]\times \cdots \times [-b_n,b_n]$ with $b_i>0$ for all $i$. For any unimodular lattice $\Lambda$ define ...
0
votes
1answer
38 views

For integers $r,s$ can $4s-1$ be a factor of $r^{2}+1$

For integers $r,s$ can $4s-1$ be a factor of $r^{2}+1$? I have found the question from here "Integer solutions for $x^2-y^3 = 23$."
4
votes
1answer
48 views

$A$ is a sum of two postive integer squares?

if $x,y,z,w$ be postive integer,and such $x^2+y^2$ is prime number,and $A=\dfrac{w^2+z^2}{x^2+y^2}\in N^{+}$ show that $A$ is a sum of two postive integer squares? maybe ...
-2
votes
0answers
32 views

Determine the splitting field of $x^n - 1$ over $\mathbb{Q}_p$ [closed]

Given a prime number $p$, how can one determine the splitting field of $x^n - 1$ over $\mathbb{Q}_p$ the p-adic number field? The case for $\mathbb{Q}$ is well known, so I am thinking that the ...
6
votes
2answers
246 views

RH would follow from $\displaystyle \frac{p_{n+1}}{p_{n+1}-1}<\frac{\log\log N_{n+1}}{\log\log N_n} $ for all $n>1$; what is my mistake?

Let $N_n=\prod_{k=1}^np_k$ be the primorial of order $n$,$\gamma$ be the Euler-Mascheroni constant and $\varphi$ denote the Euler phi function. Nicolas showed that if the Riemann Hypothesis is true, ...
0
votes
2answers
23 views

How to find all primitive triples (a,b,c)? (Pythagorean Triples)

how to find all primitive triples when one value of (a,b,c) is given? For example in this case a = 45. What is the procedure to find the primitive triples ? Conditions for primitive triples are: ...
5
votes
3answers
76 views

Find all Gaussian integers $α, β, γ$ such that $αβγ = α + β + γ = 1$

I tried to solve for this by assuming $α=a+bi$, $β=c+di$, and $γ=e+fi$, and explicitly solving this by equal $a+c+e=1$, $b+d+f=0$, and similarly for $αβγ=1$. Is there any other easier approach for ...