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
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1answer
136 views

quadratic extension of $\mathbb{Q}(X)$ generated by the square root of a square-free polynomial

Let $f(X)\in\mathbb{Q}[X]$ be a square-free polynomial, that is, not divisible by any prime of $\mathbb{Q}[X]$. Let $K:=\mathbb{Q}(X)[Y]$, where $Y$ is a square root of $f(X)$, or equivalently $Y$ ...
1
vote
1answer
105 views

Unramification and compositum

The background is: a field $K$ complete with respect to a discrete valuation $|\ |$. We write $A$ and $k$ for his discret valuation ring and the residue field of $A$. We assume that $K$ and $k$ are ...
2
votes
5answers
205 views

On prime numbers

let $q$ be a prime let $p = 2^q -1 $ is p must be prime always for any prime q ? is this is true always ? or it is false for some prime q ? if it is false , give an example to show that there ...
12
votes
1answer
273 views

primes represented integrally by a homogeneous cubic form

Expired by this question Show determinant of matrix is non-zero I am moved to ask: Given integers $a,b,c,$ and cubic form $$ f(a,b,c) = a^3+2 b^3-6 a b c+4 c^3 = \left|\begin{bmatrix} a & 2c ...
8
votes
2answers
309 views

Diophantine Quintuple?

I have come across the following set of numbers: $\{1, 3, 8, 120\}$ These are positive integers where the product of any two of the numbers equal to a number that is one less than a square number. ...
4
votes
1answer
101 views

quadratic extension of $\mathbb{Q}(X)$

Consider the ring $\mathbb{Q}[X]$ of polynomials in $X$ with coefficients in the field of rational numbers. Consider the quotient field $\mathbb{Q}(X)$ and let $K$ be the quadratic extension of ...
4
votes
2answers
238 views

Rational approximations to $\sqrt 2$

I find this problem is very interesting, but now I can't solve it. Given $n$ a positive integer, let $$f(n)=\min_{m\in\Bbb Z}{\left\lvert\sqrt{2}-\dfrac{m}{n}\right\rvert}.$$ If there is a sequence ...
3
votes
5answers
995 views

How to create a generating function / closed form from this recurrence?

Let $f_n$ = $f_{n-1} + n + 6$ where $f_0 = 0$. I know $f_n = \frac{n^2+13n}{2}$ but I want to pretend I don't know this. How do I correctly turn this into a generating function / derive the closed ...
2
votes
2answers
112 views

Solve for system of diophantine equations

$\cases{x+1=a^2 \cr x^3-x^2+1=b^2}$ I just can found a trivial solution $x=0$. Is there any other ?
1
vote
6answers
563 views

power set of natural numbers equal to the power set of integers

I need show that the two given sets: power set of natural numbers and power set of integers, have equal cardinality by describing a bijection from one to the other (describe the bijection with ...
1
vote
2answers
62 views

How to find $a,b\in\mathbb{N}$ such that $c = \frac{(a+b)(a+b+1)}{2} + b$ for a given $c\in\mathbb{N}$

Suppsoe that $$c = \frac{(a+b)(a+b+1)}{2} + b$$ Now $c$ is given - how does one find satisfying $a, b$?
7
votes
1answer
448 views

Prime number generating function as product expansion

I am interested in prime number generating function. $$f(x)=1+\sum \limits_{k=1}^\infty p_{k}x^k=1+2x+3x^2+5x^3+7x^4+11x^5+....$$ I would like to find the function as product expansion and to check ...
0
votes
2answers
28 views

For what range does this floor function scale to?

I have $\lfloor\frac{X}{(2y+1)^2}\rfloor = k$ where $X$ and $k$ are known. For what values of $y$ will this hold true? edit: all are positive integers
3
votes
1answer
50 views

Representing an element mod $n$ as a product of two primes

Given a positive integer $n$ and $x \in (\mathbb{Z}/n\mathbb{Z})^*$ what is the most efficient way to find primes $q_1,q_2$ st $$q_1q_2 \equiv x \bmod n$$ when $n$ is large? One option is just to ...
1
vote
1answer
165 views

Are there any errors in my proof that only perfect squares have rational square roots?

This is a very simple proof, but I know that proofs which are this simple can often have some erroneous assumptions. Is mine okay? Argument: With the exception of perfect squares, there are no ...
0
votes
2answers
219 views

Solve for diophantine equation $x^n + y^n + z^n =1$ [closed]

Solve for diophantine equation $x^n + y^n + z^n =1$ $x^n+y^n+z^n=2$ Is this equation solve-able ?
11
votes
3answers
355 views

Problem with infinite product using iterating of a function: $ \exp(x) = x \cdot f^{\circ 1}(x)\cdot f^{\circ 2}(x) \cdot \ldots $

[update]: I made the question more precise, more general and added a follow up question Considering the iteration of functions (with focus on the iterated exponentiation) I'm looking, ...
1
vote
2answers
155 views

Congruence equation mod $p$ involving the multiplicative order

Say $p$ is an odd prime s.t $p$ doesn't divide $x$. Let $x$ belong to the exponent $n$ modulo $p$. I need to show that if $n>1$, then $x + x^2 + ... + x^{n-1} ≡ -1 \mod p$ I'm not sure how to go ...
2
votes
3answers
2k views

How to find all the primitive roots in $\mathbb{Z}/49\mathbb{Z}$.

I need to find all the primitive roots of 49. First note, $ ϕ(49) = 42 $ Is there an easier way to go about trying all numbers less than $42$ to find the primitive roots of $49$ if we already know ...
1
vote
4answers
77 views

Simple Modulo Questions

Hey guys I have some questions regarding modulo. Some of these are solvable and some are not but I have to prove why they have no solution. Any help would be appreciated thanks! $39x\equiv65 \pmod ...
1
vote
1answer
76 views

Something like an incomplete gamma function

I want to compute $\int_0^z t^{-b}e^t \,dt$ where $b>0$ by using incomplete gamma function. Can I rewrite my integral as a form of the incomplete gamma function?
9
votes
3answers
674 views

A cohomological statement equivalent to the Riemann Hypothesis

Is there a possibility for looking for a theory of cohomology and an equivalent cohomological statement for Riemann hypothesis over $\mathbb{Z}$?
1
vote
1answer
159 views

Dilogarithm Identities

Is there a cleaner way to write: $$ f(x) = \operatorname{Li}_2(i x) - \operatorname{Li}_2(-i x) $$ in terms of simpler functions? I don't know enough about dilogarithms, and the basic identities I see ...
1
vote
0answers
55 views

Embedding an $n$-simplex in $\mathbb{Z}^n$.

I am trying to understand the proof of embedding an $n$-simplex in $\mathbb{Z}^n$ for specific values of $n$. The proof can be found here. I am stuck on what is meant by "the reflection with axis ...
2
votes
0answers
67 views

If $x \sim U(Z_n^*)$ then $x^2 \pmod n\sim U(QR_n)$?

Define: $Z_n^*=\{x \in Z_n | \operatorname{gcd}(x,n)=1\}$ $QR_n=\{x \in Z_n | \exists r \in Z_n \; s.t. \; r^2 =x\}$ How can I show that $x \sim U(Z_n^*) \implies x^2 \pmod n \sim U(QR_n)$? Thank ...
2
votes
1answer
1k views

How to calculate $ 1^k+2^k+3^k+\cdots+N^k $ with given values of $N$ and $k$? [duplicate]

Here $ 1<N<10^9$ and $0<k<50$ So we have to calculate it in order of $O(\log N)$.
7
votes
2answers
247 views

Bounds on a sum of gcd's

Does there exist a positive real number $C$ and a positive integer $M$ such that for all $n > M$ we have: $$\sum_{i=1}^n\sum_{j=1}^n\gcd (i, j)\ge Cn^2 \log n$$ This originally appeared as an ...
3
votes
4answers
139 views

Find $x$ such that $\sum_{k=1}^{2014} k^k \equiv x \pmod {10}$

Find $x$ such that $$\sum_{k=1}^{2014} k^k \equiv x \pmod {10}$$ I knew the answer was $3$.
8
votes
3answers
486 views

Proof of Wolstenholme's theorem.?

According to the theorem : $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...+\frac{1}{p-1} =\frac{r}{q}$$ And we have to prove that $r= 0 \pmod{p^2}$. (Given $ p>3$, ...
1
vote
2answers
62 views

Integral extensions

Let $p\neq1$ be an integer and let $\beta$ be a root of $x^6-p$. What is the difference, in terms of $\mathbb{Z}$-modules, between $\mathbb{Z}[\beta]$ and $\mathbb{Z}[\beta^2,\beta^3]$? I can ...
3
votes
2answers
543 views

Questions regarding p-adic expansion and numbers

As opposed to real number expansions which extend to the right as sums of ever smaller, increasingly negative powers of the base $p$, $p$-adic numbers may expand to the left forever, a property ...
40
votes
5answers
2k views

Intuition for the Importance of Modular Forms

I am learning about modular forms for the first time this term and am just starting to wrap my head around what might be the big picture of things. I was wondering if the following interpretation of ...
11
votes
4answers
869 views

Fermat's 'proof' of his Last Theorem

The definition of a unique factorisation domain came up in my rings lecture about a week ago, and my lecturer mentioned that Fermat's 'proof' of his Last Theorem probably relied on the (false) ...
0
votes
2answers
175 views

Is the Copeland–Erdős constant a random number? How is it normal?

The Champernowne constant is not random. Is the Copeland–Erdős constant random? Also if Copeland–Erdős number is normal, then shouldnt the number of $5$s and even digits be low because they cannot ...
1
vote
1answer
173 views

Type of periodicity in champernowne constant.

Digits of Champernowne constant are aperiodic, else it will be rational. Fine! But it is not random because I can write a program which will give me the position of every digit. E.g. I can calculate ...
7
votes
1answer
125 views

Solve $x^{2k}+(x-2)^{2k}=2k$ with $k\in \mathbb N$ and $x\in\mathbb{R}^+$

Solve $x^{2k}+(x-2)^{2k}=2k$ with $k\in \mathbb N$ and $x\in\mathbb{R}^+$. I have no idea how to solve this, I just can find one solution, $x=k=1$.
12
votes
2answers
221 views

Find $x\in \mathbb{Z}$ such that $54x^3+1$ is a cube

Find $x\in \mathbb{Z}$ such that $54x^3+1$ is a cube. I found $x=0$, any others ?
12
votes
2answers
389 views

Can the Basel problem be solved by Leibniz today?

It is well known that Leibniz derived the series $$\begin{align} \frac{\pi}{4}&=\sum_{i=0}^\infty \frac{(-1)^i}{2i+1},\tag{1} \end{align}$$ but apparently he did not prove that $$\begin{align} ...
11
votes
1answer
189 views

Orientation on finite dimensional vector spaces over finite fields.

For finite-dimensional $\mathbb R$-vector spaces, we define an orientation to be an equivalence class of ordered bases, where $B_1 \sim B_2$ iff the change of basis matrix $A$ taking $B_{1}$ to ...
2
votes
0answers
64 views

Lower bound on diophantine system of inequalities with all but one non-linear constraint

I have a system of $n+1$ diophantine inequalities, in the following form: $$f_{1}(x_1, x_2, \dots, x_n) \geq 0$$ $$f_{2}(x_1, x_2, \dots, x_n) \geq 0$$ $$\vdots$$ $$f_{m}(x_1, x_2, \dots, x_n) ...
1
vote
2answers
150 views

Please help me to find the value of $ABCDE$ (step by step)

$ABCD\times E = DCBA$ with $A,B,C,D$, and $E$ distinct decimal digits (and $ABCD$ representing the concatenation of those digits). How can I find the value each of them?
7
votes
3answers
392 views

Using Hensel's Lemma to Factor a Polynomial over $\mathbb{Z}_4[x]$

We recently learned about codes over $\mathbb{Z}_4$, and Hensel's Lemma. The lemma is as follows: Let $f(x) \in \mathbb{Z}_4[x]$. Suppose $\mu(f(x)) = h_1(x)h_2(x) \cdots h_k(x)$, where $h_1(x), ...
1
vote
0answers
353 views

How to prove a generalized Gauss sum formula

I read the wikipedia article on quadratic Gauss sum. link First let me write a definition of a generalized Gauss sum. Let $G(a, c)= \sum_{n=0}^{c-1}\exp (\frac{an^2}{c})$, where $a$ and $c$ are ...
1
vote
0answers
107 views

Efficient way for computation of derivatives of $f(x) = \zeta(1-x) + 1/x $ at integer x?

For some exercises with (divergent) summation of the Stieltjes constants I'm trying a formula, which involves derivatives of the $\zeta()$ -function at negative integers; perhaps better formulated as ...
0
votes
1answer
94 views

Prime Number Theorem

Using the prime number theorem, show that: $\vartheta (x) \sim x$ Where $\vartheta (x) := \sum_{p \le x} \log p$ Any help on this would be great, thanks in advance.
1
vote
1answer
641 views

Set of numbers pairwise relatively prime

Given a positve integer n, we can find infinitely many positve integers $b$ such that the $n-1$ integers in the set $\{b+1,\,2b+1,\,3b+1,\,...,\,(n-1)b+1\}$ are pairwise relatively prime. I assume ...
2
votes
2answers
167 views

Stirling's Approximation

A sharp Stirling's approximation form states that $$n! \sim \left(\frac{n}{e}\right)^n\sqrt{2\pi n}.$$ Use that form to show that: $$\binom{2m}{m} = \Theta\left(\frac{2^{2m}}{\sqrt{m}}\right).$$
0
votes
1answer
257 views

Euler's Phi Function Worst Case

For each positive integer $k$, let $P_k$ denote the product of the first k primes. Show that $\varphi(P_k) = \theta(P_k / \log \log P_k)$ is the worst case, in the sense that $\varphi(n) = \Omega(n / ...
1
vote
2answers
72 views

Sequences and Languages

Let $U$ be the following language. A string $s$ is in $U$ if it can be written as: $s = 1^{a_1}01^{a_2}0 ... 1^{a_n}01^b$, where $a_1,..., a_n$ are positive integers such that there is a 0-1 ...
1
vote
1answer
83 views

Proof of Generalized Primorial Primes

Let's call the numbers of the form $k\times p\# \mp1$, the Generalized Primorial Primes. One can find many $k$ for a fixed $p$ such that $k\times p\# \mp1$ be prime. As an example for $p = 8933$ ...