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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7
votes
1answer
481 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
29 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
173 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
222 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
358 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
158 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
78 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
735 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
175 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
58 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
259 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 $3$.
7
votes
3answers
597 views

Proof of Wolstenholme's theorem

According to the theorem, if $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\cdots+\frac{1}{p-1} =\frac{r}{q}$$ then we have to prove that $r\equiv0 \pmod{p^2}$. (Given $p>3$, otherwise ...
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
600 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 ...
41
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 ...
12
votes
4answers
940 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
187 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
176 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
222 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 ?
14
votes
2answers
410 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
201 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
156 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
426 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
1answer
383 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
110 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 ...
-1
votes
1answer
111 views

The asymptotic of the first Chebyshev function, using the Prime Number Theorem [closed]

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
669 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
171 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
270 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
73 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
90 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 $p = 8933$ ...
7
votes
2answers
116 views

Solutions of the Diophantine equation $x^2(x^2+10)=3y^2(y^2+10)$

I am looking for the solutions of the Diophantine equation $$x^2(x^2+10)=3y^2(y^2+10).$$ Is there any solution of this equation except when $(x,y)=(0,0)$? Or Any computer programme such as MAGMA ...
7
votes
1answer
151 views

modulo of sums of consective powers

I am thinking of whether there is any pattern about sums of consective powers mod $m$. Assume $m$,$n$,$k$ are integers. Denote $$f_k(n)=1^k+2^k+\cdots+n^k,$$ The question is: how does $f_k(n)$ ...
10
votes
1answer
210 views

Elementary proof that $\mathfrak{p}$ unramified in $L,L'$ implies unramified in $LL'$?

Let $K$ be a number field, let $\mathfrak{p}$ be a prime of $K$, and let $L,L'$ be extensions of $K$. Suppose $\mathfrak{p}$ doesn't ramify in either $L$ or $L'$. Is there a simple proof that ...
0
votes
2answers
53 views

is there an analytic solution to $n^2+kn-d=m^2$ m,n integers

For $k=24,d=-17;m=8,n=3$, completing the square gives $(12+n)^2=m^2+161$ Where $161$ just happens to be the product of two primes $(q=7,p=23)$, so for large $k,m,n$ factoring may be very slow. ...
5
votes
2answers
299 views

Sum of squares of sum of squares function $r_2(n)$

Let $r_2(n)$ denote the number of representations of $n$ as a sum of two squares. What is known about the sum of squares of this function, $\sum_{i=1}^n r_2(i)^2$ In particular is anything ...
5
votes
1answer
214 views

A Gauss sum like summation

I would like to calculate the following sum. Let $\zeta$ be a primitive $n$ th root of unity for some integer $n$. Here $n$ is not necessarily prime. The sum is $$\sum_{j=1}^n (-1)^j ...
2
votes
3answers
132 views

Question about divisibility of the factorial

I've found this question in the book devoted to Lie's groups. How to prove that $n!$ is devided by $n_{1}!n_{2}!...n_{k}!$, where $n=n_{1}+n_{2}+...+n_{k}$ and $n$, $n_{i}$ are natural numbers?
6
votes
1answer
341 views

A question on Paul Erdős's research on Egyptian fractions

A good day to everyone! I have a (somewhat) intriguing question regarding Paul Erdős's papers on Egyptian fractions (e.g., his 2nd paper during his mathematical career was about this topic). My ...
2
votes
2answers
154 views

Prove that the sequence satisfying $a^{p^n}\equiv a^{p^{n-1}}$mod $p^n$ for $n\geq 1$, converges to the Teichmuller Representative

Prove that for any $a\in\mathbb{Z}_p$, the sequence satisfying $a^{p^n}\equiv a^{p^{n-1}}$mod $p^n$ for $n\geq 1$, converges to the Teichmüller Representative congruent to $a$ mod $p$. So a p-adic ...
3
votes
1answer
83 views

Prove that the following sum is 1.

I should prove this: $$\sum_{c_1,c_2,...,c_n} \dfrac{1}{c_1!c_2!\cdot\cdot\cdot c_n! 1^{c_1}2^{c_2}\cdot\cdot\cdot n^{c_n}} = 1$$ where $c_1+2c_2+...+nc_n = n$ (1) Evidently, the summation is all ...
14
votes
1answer
301 views

Why is $x^3-5x$ injective on the rationals?

I've found the statement on the internet that the polynomial $x^3-5x$ is injective on the rational numbers, but without any comments on how to prove it. I think it means it must be easy, but I don't ...