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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2
votes
1answer
32 views

Understanding Wright's proof of Landau's theorem

I'm reading Wright's A simple proof of a theorem of Landau in which the core argument is a proof by induction and I find myself stuck on a major point. I must be misunderstanding notation or something ...
0
votes
0answers
23 views

Prove that there exists a prime numbers $p$ such that $\min(S_{p}(a)+S_{p}(n-a),S_{p}(b)+S_{p}(n-b))\ge p-1+S_{p}(n)$

Let $n$be an integer greater than $1$ and let $a,b$ be postive integers smaller than $n$, Prove that there exists a prime numbers $p$ such that $$\min(S_{p}(a)+S_{p}(n-a),S_{p}(b)+S_{p}(n-b))\ge ...
6
votes
0answers
71 views

Prove that for each integer $n \ge 2$ there exists a prime number $p$ dividing $a_n$

Let $c \ge 1$ be an integer. Define a sequence of positive integers by $a_1 = c$ and $$a_{n+1}=a_n^3-4c\cdot a_n^2+5c^2\cdot a_n+c$$ for all $n\ge 1$. Prove that for each integer $n \ge 2$ there ...
-1
votes
0answers
76 views

Does 0.9 periodic exist as a number [duplicate]

If I understand correctly periodic numbers are representations of rational numbers. For instance $1/3$ is $0.\overline 3$ periodic. Also I understand that one can always construct a rational number ...
-3
votes
1answer
31 views

A congruence equation [duplicate]

I don't have any clue . Can chinese remainder theorem can be used? I am studying number theory but I am not an expert as you can see. This question was set in the exam and ...
5
votes
1answer
32 views

Prove that there exists infinitely many positive integers $n$,such $nS_{p}(n)+2014n\ge S_{p}(n^2)+2014S_{p}(n)$

Prove that there exists infinitely many positive integers $n$ such $$nS_{p}(n)+2014S_{p}(n)\ge S_{p}(n^2)+2014n$$ or $$(n+2014)S_{p}(n)\ge S_{p}(n^2)+2014\cdot n$$ Here $S_{p}(n)$ is ...
2
votes
1answer
44 views

A question from Titchmarsh's zeta function book.

On page 30, he writes that $\xi(0)=-\zeta(0)=1/2$, but on page 16 he writes that: $\xi(s)=1/2 s(s-1)\pi^{-1/2s}\Gamma(1/2s)\zeta(s)$ in eq.(2.1.12); so if I plug into this equation $s=0$ then I get ...
0
votes
0answers
27 views

Unique integer solutions for $\frac{1}{x} + \frac{1}{y} = \frac{1}{z} $, with $x,y,z \ne 0 $, given $z$ [duplicate]

What are the unique integer solutions for a given integer $z$ for $\frac{1}{x} + \frac{1}{y} = \frac{1}{z} $, with $x,y,z \ne 0 $? From what I can tell, $x|yz,\ y|xz,\ z|xy$, so $x,y,z$ must ...
2
votes
3answers
92 views

Can $x\pi$ be rational?

When I was solving a math test, I came across this problem - Let $x$ be an irrational number. What type of number is $x\pi$? a) Rational only b) Irrational only c) Could be rational ...
5
votes
3answers
117 views

Generalize multiples of $999…9$ using digits $(0,1,2)$

The smallest $n$ such that $9n$ uses only the three digits $(0,1,2)$ is $1358$, giving a product $12222$. For $99n$ this is $11335578$, giving $1122222222$. Similarly, ...
1
vote
3answers
56 views

Estimating the sum of reciprocals of products of two primes

It's rather well-known that $$ \sum_{p \leq X} \frac{1}{p} \sim \log \log X,$$ where this is a sum over the positive integer primes. Can we efficiently estimate the sum $$ \sum_{p,q \leq X} ...
1
vote
2answers
41 views

Doubt whether proof is correct

I'm not very good at number theory, so I would rather ask. The proposition states that for every even square $x$ there is an odd square $y$ such that $x+y$ is a square number. The proposition comes ...
2
votes
1answer
35 views

a function as an infinite series of e

My literature is rather lacking, and I can't seem to find one that tackles this kind of infinite series: $$ f(x)= \sum_{n=1}^\infty e^{g(n)x} $$ The goal is to find $g(n)$ given $f(x)$. What area ...
0
votes
1answer
31 views

Upper bound of digit sum of powers

Take $x \in \Bbb N$, $x \le9$ and $m \in \Bbb N$. Now we define a function $d_s(n): \Bbb N \to \Bbb N$ as the digit sum of $n$ in base $10$. Now let's say we have a lower bound $b_l$ and an upper ...
1
vote
4answers
75 views

Prove that if $a^2+b^2$ is a multiple of three, then a and b are multiples of three

I have attempted to prove the above. I am uncertain about the correctness of my proof: Both numbers have to be multiples of three, i.e. $3a+3b=3n$, $\ 3(a+b)=3n$ It is not possible to arrive at an ...
-1
votes
1answer
52 views

euler function property $\varphi(n)\geq\sqrt{n}$ [duplicate]

How to prove that $\varphi(n)\geq \sqrt{n}$ for every positive integer $n$ distinct of $2$ and $6$.
0
votes
0answers
29 views

A Curious Chinese Remaindering Question

Given integers $A,B,N$ with $0<N<A,B<N(4\log N)^{1+\delta}$ with $\delta>0$ arbitrarily small, do there exist integers $X,Y$ such that $$0<X<A,B<X\log ...
4
votes
3answers
785 views

The truth of Nyonyon Theorem

Let $s= ab$ be a semiprime number. Then the Nyonyon Theorem states that $s+a$, $s+b$, $s+a+b$ are not all coprime to three. (In other words: there exist no $s= ab$ semiprimes such that $s+a$, $s+b$ ...
1
vote
2answers
40 views

Find all natural values n, that $\sqrt{P_{2}(n)}$ is also a natural number

I have a polynomial of the second degree $a\cdot n^2 + b \cdot n + c$ and I need to find out natural numbers $n$, such that $\sqrt{a\cdot n^2 + b \cdot n + c}$ is also a natural number. After ...
0
votes
3answers
108 views

Find all positive integers for the following question [closed]

Find all positive integers that makes the result of $$\frac{1}{x}+\frac{1}{y}$$ integers
3
votes
0answers
47 views

Linear independence of primitive Dirichlet characters and convolution

This is not an exercise but merely a question I have. Fix $N \in \mathbb{N}$ and suppose there exist some values $a_k \in \mathbb{C}$, for $k \in \mathbb{Z}_N$, such that $$ \sum_{k \in \mathbb{Z}_N} ...
2
votes
1answer
143 views

Can I have a logical explanation for why this number is so ridiculously close to a whole number? [duplicate]

$$e^ {\pi\sqrt{163}}=262537412640768743.9999999999992\cdots$$ Why does this number run so incredibly close to a whole number? Can I have a logical explanation for why this finding? I know how to ...
0
votes
2answers
60 views

Books on number theory

I had a very short introduction to number theory in one of the classes I took and I learned a bit about divisibility and congruence, not much further than Fermat's little theorem, and I would like to ...
2
votes
3answers
34 views

Question on a passage from “Rational Points on Elliptic Curves”

I was reading the book "Rational Points on Elliptic Curves", when I've crossed with the following passage: "(...) since $3$ does not divide the order $p-1$ (where $p$ is a prime) of the cyclic group ...
1
vote
3answers
64 views

Help with proof of $(n+1)^n > n! 2^n$

I have already managed to prove it using induction and Bernoulli's inequality but I wonder if there is another way. My proof goes like this: (This is my first time using MathJax, so I apologize for ...
-3
votes
1answer
48 views

Example of a diophantine polynomial

A diophantine set is a subset of a power $\mathbb{Z}^k$ of the set $\mathbb{Z}$ of integers which can be written as $$\{x \in \mathbb{Z}^k : \exists y \in \mathbb{Z}^m : P(x, y)=0\}$$ where $P$ is a ...
-1
votes
1answer
38 views

Solution verification regarding several questions on number theory? [closed]

1) List all the integers between $100$ and $300$ which are $11(mod \ 17)$ Solution: Let $a$ be the required number. So $a \in (100,300)$. And $a \equiv 11(mod \ 17)$. Or $a=11+17k$ for any integer ...
3
votes
1answer
38 views

Inverse problem can't solve it,:$2^{m+1}+1\mid 3^{2^{m}}+1$,when$2^{m+1}+1$ is prime number,

Let $m$ be a possitive natural number, and $2^{m+1}+1$ is prime number, show that:$2^{m+1}+1\mid 3^{2^{m}}+1$. This question inverse problem I can solve it, see this linksmathlinks
4
votes
2answers
47 views

A positive integer is equal to the sum of digits of a multiple of itself.

Let $n$ be a positive integer, prove there is a positive integer $k$ so that $n$ is equal to the sum of digits of $nk$. I'm not really sure how I should approach this problem, I tried to do a ...
1
vote
0answers
26 views

How to prove that sums of even powers is divisible by p

For $n\leq (p-2)$ I want to prove that $\sum_{k=0}^{p-1} (r+k)^{n} \equiv 0 \pmod{p}$ It is easy to see that it is true for odd n, since $(-a)^k \equiv -a^k$, and you can just pair up terms since ...
4
votes
4answers
51 views

How many four digit numbers are perfect square whose first and last two digits are same?

I tried it by assuming the number as $\sqrt{1100a+11b}$ and than tried to find figure out perfect square but I am unable to approach further.
2
votes
2answers
47 views

Ways to make change

Given unlimited coins with values $1^2$, $2^2$, $3^2$, $4^2$,..., $17^2$ Now given an amount X, in how many ways can we exchange it using these coins? Example for $X=24$ answer is $16$. It means ...
8
votes
1answer
73 views

Zeta function, $\mathbb{F}_5[T, \sqrt{T(T-1)(T+1)}]$

Let $A = \mathbb{F}_5[T, \sqrt{T(T-1)(T+1)}]$. My question is, what is the easiest way to see that$$\zeta_A(s) = {{1 + 2 \cdot 5^{-s} + 5^{1 - 2s}}\over{1 - 5^{1 - s}}}?$$Much thanks in advance. ...
0
votes
1answer
47 views

How many times does a given prime occur as factor below a number?

Given an integer $N>0$, a prime $p>2$, what is closed form of the sum $$\large\sum_{\substack{3\,\leq\, m\,\leq\, N\\p^i\mid m\strut\\{m\bmod p^{i+1}}\,\neq\,0}}i$$ which is ...
0
votes
2answers
52 views

Analytic continuation of primality function

(There is my initial question, but by advice of @Charles I'm splitting it) For integers we have a primality function: $$ isprime(n)=\begin{cases}1,&\text{$n$ is prime}\\0,&\text{$n$ is not ...
2
votes
1answer
21 views

If $f,g$ in $Z[x]$, $h$ in $R[x]$ with $f=gh$, is $h$ nessecarily in $Z[x]$?

Let $f$ and $g$ be monic polynomials in $Z[x]$. There exists a polynomial $h$ in $R[x]$ such that $f=gh$ for all real $x$. Is $h$ nessecarily in $Z[x]$?
3
votes
0answers
20 views

$O_S$ is the integral closure of $k[T]$ in $F$ for some embedding of $k(T)$ in $F$?

Let $F$ be a function field in one variable over a field $k$. Let $S$ be a nonempty finite subset of the set of all places of $F$. Let$$O_S = \{f \in F: \text{ord}_v(f) \ge 0 \text{ for all }X ...
0
votes
0answers
17 views

Number of solutions of the following congruence

How do I find the number of solutions of the congruence $z^2=d(\mod 4n)$. d is the discriminant of the of the quadratic field equation $ax^2+bxy+cy^2$. I am trying to evaluate the number of terms of ...
0
votes
1answer
30 views

For $n$ an even number and $p$ a prime, does $\lfloor {\frac {n}{4}} \rfloor= \frac{1}{\pi (n)-\pi (n/2)} \sum_{n/2<p<n} p-\frac {n}{2}$ hold?

I was playing around with prime numbers and I noticed that for $n$ an even number, the average of the distance between all primes between $n/2$ and $n$ and $n/2$ is equal to $\lfloor {\frac {n}{4}} ...
2
votes
0answers
30 views

Calculating the Galois group of a covering map

Suppose $C$ is an algebraic curve and $\phi:C\rightarrow \mathbb{P}^{1}$ is a covering map of the complex projective line ramified at $\{0,1,\infty\}$ only. Suppose $\phi':C'\rightarrow ...
4
votes
6answers
155 views

Why does the $\sum_{n=1}^x \frac {1}{n} \sim \mathrm {ln} x + \frac {1}{2}$

I was playing around with the Harmonic Series and I noticed that: $$\sum_{n=1}^x \frac {1}{n} \sim \mathrm {ln} x + \frac {1}{2}$$ I wanted to know if this is just some coincidence or if it is caused ...
2
votes
1answer
56 views

Solutions of $(2x-1)^x\equiv1\mod\ p$ [closed]

Has the equation $(2x-1)^x\equiv 1\mod{p}$, for $p=1+6qx$, where $p$, $q$ are primes, $x$ is an odd integer and $x<p$ any solutions except $x=1$? Many thanks.
13
votes
1answer
132 views

“Binomiable” numbers

Is there a nice criterion to determine whether a given natural $m$ can be written as a binomial number $\binom{n}{k}$ with $1 < k < n-1$? I've been thinking on this problem with a friend and ...
3
votes
1answer
37 views

Question regarding number congruences?

First of all, before the question, I want to clear that how does $17x \equiv 1 \pmod 4 $ imply $x \equiv 1 \pmod 4$? I did: $17x \equiv 1 \pmod 4 $ $16x \equiv 0 \pmod 4$ Subtracting both, We ...
3
votes
5answers
132 views

Why is $-i^3 = i$?

Why is the value of $-i^3$ equal to $i$? After experimenting, I got this result - $-i^3=-i^2\cdot -i=1 \cdot -i=-i$ What is the error in my proof? EDIT Here is the original proof - ...
1
vote
0answers
21 views

Holomorphicity and modularity of Jacobi forms

I am reading this paper Suerconformal algebras and mock theta functions by T. Eguchi and K. Hikami. In section 3.2, the authors define a function $$J(z;w;\tau) = ...
8
votes
2answers
79 views

Find all postive integers $n$ such $(2n+7)\mid (n!-1)$

Find all postive integers $n$ such that $$(2n+7)\mid(n!-1).$$ I have $n=1,5$, but can not find any other and can not prove whether there is any other solution or not.
0
votes
2answers
62 views

Is the integer $0$ a deficient number?

It is well known that the divisors of the integer $0$ are all non zero-integers numbers ,the sum of those divisors greater than $0$, then is it a deficient number ? Thank you for any help
6
votes
1answer
41 views

Median order of an element in an additive group modulo $n$

I'm trying to gain some intuition here. Suppose we have the group $\mathbb{Z}_{n}$ (with the operation being addition modulo $n$). What is the median order of an element of this group when $n$ is a ...
3
votes
0answers
36 views

Cokernel of map, function field.

Let $F$ be a function field in one variable with total constant field $k$, let $X$ be the set of all places of $F$, and let $S$ be a nonempty finite subset of $X$. We are interested in the dimension ...