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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22
votes
2answers
329 views

Is there an elementary proof that $\sum_{n=1}^\infty {1\over n^s\{n\pi\}}<\infty$ for some $s>0$?

Edit: David Speyer's answer made me realize a couple of things and I would like to clarify. Sorry if the length of this is getting out of hand. First, it is now clear that no estimate can be obtained ...
-2
votes
1answer
107 views

Prove that euler number_40 is a prime =)) [closed]

How can i prove that euler number_40 is a prime? Euler numbers are numbers that can be written as $E_n=n^2-n+41$ where $n$ is an integer. Now i need to prove that $E_{40}$ is prime
1
vote
1answer
50 views

Stronger condition then ultrametric condition on metric space

A metric space $(X,d)$ is called an ultrametric space if it is a metric space and fulfills the stronger triangle inequality (see Wikipedia) $$ d(x,y) \le \max\{ d(x,z), d(z, y) \}. $$ Examples are ...
1
vote
2answers
133 views

Suppose $γ$ is a $k$th root of unity that satisfies a quadratic equation $z^2−mz−n=0$ with $m,n\in\mathbb{Z}$. Then $k=3,4$ or $6$

Let $k\in\mathbb{Z}$ with $k>2$ and suppose $\gamma$ is a $k$th root of unity that satisfies a quadratic equation $z^2-mz-n=0$ with $m,n\in\mathbb{Z}$.Then $k=3,4$ or $6$. My knowledge on ...
0
votes
1answer
70 views

Problem in book Art of Problem? [closed]

Let $a$ and $b$ be positive integers, such that $a \mid b^2$, $b^2 \mid a^3$, $a^3 \mid b^4$, $b^4 \mid a^5\ldots$. Prove that $a = b$.
3
votes
1answer
166 views

Subsets of all Diophantine's sets

Function $\mathbb{N}^k \to \mathbb{N}^m$ is computable $\Leftrightarrow$ graph of function is Diophantine. Consider some subset $S$ of computable functions (for example some Grzegorczyk's class or ...
11
votes
0answers
170 views

Biggest powers NOT containing all digits.

Let $m>1$ be a natural number with $m \not\equiv 0 \pmod{10}$ Consider the powers $m^n$ , for which there is at least one digit not occurring in the decimal representation. Is there a largest $n$ ...
1
vote
0answers
82 views

Decomposition subgroup of Cyclic Galois Extension

My question: Say we have a cyclic Galois extension of degree $n$ over $\mathbb{Q}$. Denote the Galois group as $G$. If $H\leq G$, then does there exist a prime, $q$ in $\mathbb{Z} \subset \mathbb{Q}$ ...
1
vote
1answer
86 views

Compute Carmichael function and Euler’s totient function

How can I compute the Carmichael function and Euler’s totient function of 172872 ? Thank you
1
vote
1answer
25 views

Testing density with a countable family

Let $d$ denote the lower density on $\mathbb{N}$, $a>0, $ $\mathbb{N}_{a}:=\left\{ B\subset\mathbb{N}:{d}(B)\geq1-a\right\} $ and $A\subset\mathbb{N}$. If $A\cap B\neq\emptyset$ for every $B\in\...
7
votes
1answer
110 views

Prime chains with large gaps

It is well known that the gap between consecutive primes is unbounded. Is this still true for a chain of consecutive primes ? More Formally : Is the following statement true for all natural numbers ...
1
vote
0answers
113 views

Estimations for the number of prime factors, counted with multiplicity (elementary combinatorics)

If $N_k(x):=\ \mid\{n\leq x : \Omega(n)=k\}\mid$, where $\Omega(n)$ is the number of prime factors (counted with multiplicity) in $n$, I am trying to reason a crude under-estimate for large $k$ and ...
0
votes
3answers
259 views

$ax^3+by^3+cz^3=0$ and Elliptic curves

What is relation between $ax^3+by^3+cz^3=0$ and Elliptic curves?
1
vote
1answer
96 views

Check a number of form $a^b$.

Given a number $n$, what is best efficient way to check if it is of $a^b$ form for some $a,b\geqslant2$. Provided $n$ can be as large as $10^{16}$.
1
vote
1answer
78 views

How would you compute this modulo problem?

$$ (10^{18}-1)(10^{18}-1) \bmod 10^{18} $$ I am solving a programming problem and I hold my integers in 64 bit long long integers. Above is a particular case I am unable to solve. $(ab)\bmod m = (a \...
0
votes
1answer
113 views

How do modular arithmetic rules hold for modulo with composite numbers?

I know that (x*y)%p = ((x%p) * (y%p))%p holds true for a prime p. Is this equation valid when p is a composite number? How do we write this equation when p is a composite number?
2
votes
0answers
46 views

Infinite series with only two zeros at $\Re(s)=\frac12$. Why is that the case?

I was experimenting with the following series: $$\displaystyle k(s)=\sum_{n=1}^\infty \frac{1}{n^{\ln (\frac{n}{s})}}$$ I believe this is an entire function without any poles in $\mathbb{C}$ (note ...
4
votes
0answers
61 views

Automorphism group of an L-function

I define the notion of Galois class of L-functions as follows: $A$ is a Galois class of L-functions if and only if the following conditions simultaneously hold true: 1) $A$ is a subset of the ...
1
vote
0answers
95 views

Estimation of a logarithmic sum

I need to estimate the sum $$ \underset{r=2}{\overset{t}{\sum}}\left(\frac{\log\log r}{r}\right)^{2}. $$ I tried to use the Abel's partial summation, and I got $$ \frac{(\log\log t)^{2}}{t}-\int_{2}^{...
3
votes
0answers
120 views

$\frac{p^p-1}{p-1}, \frac{p^p+1}{p+1} $ cannot be prime power at the same time

$p\gt3$ is a prime, then the two numbers $$\frac{p^p-1}{p-1}, \; \frac{p^p+1}{p+1} $$ cannot be prime power at the same time I have no clue about it. Could anyone help me? Thanks a lot.
0
votes
1answer
33 views

Comparing sums and sums of products

Suppose that $\sum_{i=1}^m c_i < \sum_{i=1}^m c_i'$ and $\sum_{i=1}^m d_i < \sum_{i=1}^m d_i'$. Is it then true that $\sum_{i=1}^m c_i d_i < \sum_{i=1}^m c_i' d_i'$? If not, is it true if $...
2
votes
1answer
55 views

Jacobi symbol $\left(\frac{(n+1)/2}{n}\right)$

Let $(\frac{a}{n})_J$ be Jacobi symbol defined by \begin{equation} \left(\frac{a}{n}\right)_J=\left(\frac{a}{p_1}\right)^{e_1}\left(\frac{a}{p_2}\right)^{e_2}\cdots\left(\frac{a}{p_k}\right)^{e_k} \...
14
votes
0answers
520 views

Prime factor of $2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4$

I checked the prime factors of $$2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4 = 2^{2^{2^2}} + 3^{3^{3^3}}$$ with trial division and found non below $8*10^9$ Nevertheless, the given number has still ...
6
votes
1answer
150 views

Geometric Proof that $\mathbb{Z}[\sqrt{-3}]$ is non-Euclidean

Is there a geometric proof showing that $\mathbb{Z}[\sqrt{-3}]$ is non-Euclidean? I think this is a sketch of how to proceed. Consider the elliptical region $x^2+3y^2<1$. We can then partition ...
2
votes
0answers
101 views

What consequence would there be if $\pi$ was not normal?

It is suspected that $\pi$ is normal, that is the distribution of its digits is uniform for any base. Would any results or algorithms, especially those that rely on probabilistic methods, be different ...
1
vote
1answer
53 views

Reference request for proof of Landau's generalised PNT

Could someone please point me in the direction of a proof for Landau's asymptotic formula for k-almost primes: $$\pi_k(n) \sim \left( \frac{n}{\log n} \right) \frac{(\log\log n)^{k-1}}{(k - 1)!}$$ I ...
5
votes
1answer
205 views

A Fermat's FLT look-like for matrices.

I was wonderwing about a matrix equation (in some way similar to a very well known theorem :P). Find all positive integers $n,m$ such that there exists $X$, $Y$, $Z$ matrices $n\times n$, non-...
0
votes
1answer
232 views

Maths without irrational numbers

Has anyone imagined so far a consistent mathematical system which can do without irrational numbers? I am writing a philosophy disertation where I claim that mathematics cannot even be imagined ...
1
vote
1answer
128 views

pairwise disjoint subsets of divisors of $ n $ (maximum number)

Let $ n \in \mathbb{N} $, $ n>1 $ and $ a_1,\ldots,a_k \in \mathbb{N} $ (not necessarily distinct!) with $ a_i \mid n $ for all $ i=1,\ldots,k $ be given. Assume that $ \sum_{i=1}^k a_i = K\cdot n $...
1
vote
0answers
60 views

Proof that Hecke operators on modular forms commute

I am working on Hecke operators on modular forms and would like to prove that these commute. Specifically, I am trying to prove that $$ T_nT_m=\sum_{d\vert (n, m)} d^{k-1} T_{\frac{nm}{d^2}}=T_mT_n, $$...
15
votes
3answers
466 views

A prime of the form $38111111\ldots$

Let $z(n)$ denote the number given by $38$ followed by $n 1$'s. What is the least number $n$, such that $z(n)$ is prime ? With brute force, I checked up to $7000$ digits and did not find a prime. ...
65
votes
0answers
1k views

Gross-Zagier formulae outside of number theory

(Edit: I have asked this question on MO.) The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer ...
1
vote
1answer
373 views

what the RH equivalent for Riemann prime formula $\Pi(x)$?

Question follow the one answered already, zeros about Riemann Zeta function and some L-function Let's me try my best to make it clear on what I am asking. In his 1859 paper "On the Number of Primes ...
2
votes
3answers
37 views

a problem on system of equation in modulo classs

Let $a$, $b$, $m$ and n be integers, $m$, $n$ positive,$ am + bn = 1$. Find an integer $x$ (in terms of $a, b, m, n, p, q$) so that $$x ≡ p (\mod m)$$ $$x ≡ q (\mod n)$$ where $p$ and $q$ are given ...
2
votes
1answer
121 views

last $2$ digit and last $3$ digit in $\displaystyle 2011^{{2012}^{2013}}$

Calculation of last $2$ digit and last $3$ digit in $\displaystyle 2011^{{2012}^{2013}}$. $\bf{My\; Try}::$ for last $2$ digit:: which is same as when we divide $2011^{2012^{2013}}$ is divided by $...
7
votes
1answer
90 views

Max value of $n$ for which $3^n\mid(80!)!$

Calculate the max value of $n$ for which $(80!)!$ is divisible by $3^n$. My Attempt: The exponent of prime factor $p$ in $(n!)$ is given as $$ v_p(n!) = \left\lfloor \frac{n}{p}\right\rfloor + \...
0
votes
3answers
905 views

Congruences of the form $x^2-a \equiv 0$ (mod pq)

Problem: Let p and q be distinct primes. What is the maximum number of possible solutions to a congruence of the form $x^2-a \equiv 0$ (mod pq), where as usual we are only interested in solutions that ...
1
vote
1answer
127 views

Ramanujan Notebook Part 1 (1.16): $\sum q^{n^2} = (-q;q^2)_\infty^2(q^2;q^2)_\infty=\frac{(-q;-q)_\infty}{(q;-q)_\infty}$

I am having trouble with proving a statement in Ramanujan's Lost Notebook Part 1 (1.16). The statement is as follows: $\varphi(q)=f(q,q)=\sum_{n=-\infty}^\infty q^{n^2} = (-q;q^2)_\infty^2(q^2;q^2)_\...
11
votes
2answers
359 views

product is twice a square

For every positive integer $n$, there exists a set $S\subset \{n^2+1,n^2+2,\dotsc,(n+1)^2-1\}$, such that $$\prod_{k\in S}k=2m^2$$ for some positive integer $m$ I have no clue about it. Could ...
2
votes
0answers
86 views

Fast check If the remainder is 1

Is there any fast method to 'say' that $R = (A \mod B)$ is $1$ or $R > 1$ or $R \neq1$ or $R > k>1$ ( where $k$ is a small integer on $32$ bits) without to actually calculate the real value ...
3
votes
0answers
220 views

Prime factor of $14 \uparrow \uparrow 3 + 15\uparrow \uparrow 3$ wanted

Checking the prime factors of the numbers $$z(a,b) \ := a \uparrow \uparrow 3 \ +\ b \uparrow \uparrow 3 \ ,$$ a,b positive integers with a < b the number $$z(14,15) = 14 \uparrow \uparrow ...
1
vote
0answers
31 views

A sequence converging to the number of decompositions of $2n$ as a sum of 2 primes

For every even positive integer $n>2$ and every non-negative integer $k$, let's define the sequence $N_{k}(n)$ as follows: $$N_{k}(n):=\displaystyle{\sum_{m=3}^{n/2}\frac{1}{\Omega(m)^k}\cdot\frac{...
2
votes
1answer
63 views

Determination of all prime numbers which give integer solution of a particular summation.

Determine all primes numbers $p$ such that $$p \sum_{k=0}^{n}\frac{1}{2k+1} \in N$$ for a given positive number $n$
3
votes
2answers
108 views

how many solutions of $ 3^a-7^b-1=0 $ equations

I want to know how many solutions solutions of $ 3^a-7^b-1=0 $ exists . I have got $ x-y=1 $ eqn when I have assumed that $3^a=x$ and $ 7^b=y $ .I cant proceed from here . Here $ a $ and $ b $ are ...
1
vote
1answer
94 views

Quadratic residues and kernel of a homomorphism

Show that if $p\equiv 3 \pmod 4$ is a prime, exactly one between $2$ and $-2$ is a quadratic residue modulo $p$. The "most obvious" solution is the following: since $\displaystyle \left(\frac{-1}{p} \...
3
votes
1answer
173 views

Bound of the sum $\sum_{p\le n}\frac{1}{\log(p)}$

While doing a sum I came to the sum $\displaystyle\sum_{p\le n}\dfrac{1}{\log(p)}$. Where the $\log$ is the natural logarithm. It was easy to prove that $\displaystyle\sum_{p\le n}\dfrac{1}{\log(p)}&...
1
vote
1answer
39 views

to show $\exists$ t$\in$ $\mathbb{Z}$ such that q=$t^2$

Suppose a,b,c $\in$ $\mathbb{Z}$ and $x=p/q$, $y=r/s$ be two rationals satisfying the equation $y^2$=$x^3$$+$$ax^2$$+$$bx$$+$$c$. Prove, $\exists$ $t$ $\in$ $\mathbb{Z}$ such that, $q$=$t^2$, $s$=$t^...
1
vote
1answer
571 views

Showing every even number can be factored as a product of E-primes

Note this problem is about the investigation of the E-Zone. Part A: Describe all E-primes. Answer: E-primes are greater than and equal to 1 and its only E-zone factors are 1 and itself (similar to ...
5
votes
3answers
133 views

Prove that any two numbers of the form $2^{2^n}+1$ are coprime to one another.

Full problem statement: Prove that any two numbers of the follwing sequence are relatively prime: $2 + 1, 2^2+1, 2^4 + 1, 2^8+1, ... 2^{2^n} + 1 $ So far I have tried to use Euclid's algorithm with ...
4
votes
0answers
149 views

Transcendental numbers involving primes?

Is the prime zeta function value $$ P(2)=\sum_{p \in \mathrm{primes}} \frac{1}{p^2} = 0.452247420041065498506543364832247934173231343\ldots $$ a transcendental number ? What about the following sum ...