Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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0answers
30 views

The diophantine equation $A^2+B^2=C^2$ for integer-valued polynomials

How can I find the solutions to this diophantine equation in $\Bbb{Z}[X]$: $$A^2+B^2=C^2 \, ?$$ Here $A$, $B$, $C$ are polynomials.
0
votes
1answer
16 views

Equivalence for binary quadratic forms with positive square discriminant

I recently encountered an interesting proposition without proof: If $f(x,y)$ is a quadratic form whose discriminant is a non-zero perfect square, then $f(x,y)$ is equivalent to a form $a*x^{2} + ...
1
vote
2answers
23 views

Stuck with modular arithmetic problem using multiplication property

I have the following problem: Given $k\geq 1$, find $h$ such that $$2^h \frac{4^k-1}{3}-1 \equiv 0 ~(\text{mod}~3).$$ This is my attempt using the invariance of multiplication: $$2^h ...
0
votes
1answer
46 views

Trying to show that $\phi(n) > c_1 \frac{n}{\text{log log }n}$ for some constant $c_1 > 0$

Trying to show that $\phi(n) > c_1 \frac{n}{\text{log log }n}$ for some constant $c_1 > 0$ where $\phi(n)$ is the euler phi function. I was wondering if I could use something like ...
1
vote
5answers
40 views

Intuition to why average of the square of a positive integer and the integer itself is the sum of all numbers from 1 to the integer?

The sum of all numbers from 1 to n, i.e. $\sum_{i=1}^n i = \frac{n(n+1)}{2} = \frac{n^2 + n}{2}$ This happens to be show that the average of a number and its square equals the sum of all numbers ...
0
votes
1answer
32 views

how many people are at the party with such restriction applied [closed]

At a party, each person shakes hands with 5 other people. There are a total of 60 handshakes. How many people are at the party? i try firs by dividing the total number of hand shake by the number of ...
1
vote
1answer
46 views

Show that if x,y,z are not divisible by 53, then $x^{26}+4y^{26} \neq\ z^{26}$

Show that if x,y,z are not divisible by 53, then $x^{26}+4y^{26} \neq\ z^{26}$ I've got that $x,y,z$ to the 52nd power are congruent to 1 modulo 53 from Fermat's. How is it continued? Help would be ...
0
votes
1answer
21 views

Find all $(h,k)$ such that $2^h \equiv 1 ~(\text{mod}~ 3^k) $

I'm facing with the following problem: Find all $(h,k)$ such that $$2^h \equiv 1 ~(\text{mod}~ 3^k) ~~~~~~~~(1)$$ and $$2^h \geq 3^k+1 ~~~~~~~~(2).$$ I'm just able to prove that the $(1)$ holds ...
-1
votes
0answers
24 views

Ways of representing the product of N numbers as sum of two squares

Given N numbers, we need to tell the number of ways of representing the product of these N numbers as sum of two squares. Example : Let $N=3$ and numbers be $[2,1,2]$ then as $2*1*2=4$ There are 4 ...
0
votes
1answer
52 views

Search for very large prime (greater than $2^{57885161} − 1$) between Crystal Numbers

Denote $p[i]$ as the $i$th prime. In my opinion, the following is true: Prime Gap Axiom There are always distinct prime factors for $\{p[i],p[i]+1,p[i]+2, \dots , p[i+1]\}$. Question 1 How to ...
2
votes
1answer
33 views

if this divisors such $d_{1}+d_{2}+\cdots+d_{k-1}=n-1$,then there exsit $m$ such $n=2^m$

Interesting Question: let $n\ge 2$ be a positive integer,with divisors $$1=d_{1}<d_{2}<\cdots<d_{k-1}<d_{k}=n$$ and such $$d_{1}+d_{2}+\cdots+d_{k-1}=n-1$$ show that:there ...
2
votes
1answer
30 views

Definition of field of fractions of $p$-adic integers

I am given this definition of the field of fractions of the p-adic integers: $$Q_p=\left\{ \frac{r}{s} \mid r, s \in Z_p, s \neq 0\right\}$$ How can I show that: $Q_p$ consists of the sums of the ...
0
votes
1answer
55 views

how many people are at the party

At a party, each person shakes hands with 5 other people. There are a total of 60 handshakes. How many people are at the party? i am lost because of the 60 hand shake that is mentioned.
0
votes
0answers
19 views

Calculate sum of distinct pairs [closed]

Given an array A we need to find the sum of all distinct pairs of indexes from the array and adds the value ⌊$A[i]+A[j]\over A[i]×A[j]$⌋ to the sum Note: ⌊$A\over B$⌋ is the integer division ...
6
votes
1answer
68 views

Most elementary proof that a determinant is divisible by $m$

So a challenge problem states that you have an $n \times n$ matrix, where each entry is an integer between $0$ and $9$, and when each row is read as a base-10 number the number is divisible by a ...
0
votes
0answers
13 views

Finding out LCMs for all possible subset. [duplicate]

The non-empty subsets of A={27,42,30,94} are {27}, {42}, {30}, {94}, {27,42}, {27,30}, {27,94}, …, {27,42,94}, {42,30,94}, {27,42,30,94}. The LCMs (least common multiples) of all these subsets are ...
3
votes
4answers
101 views

Prove that $2^{10}+5^{12}$ is composite

Prove that $2^{10}+5^{12}$ is composite I need to solve this using only high school mathematics. Any ideas?
0
votes
1answer
30 views

Finding out LCM(least common multiple)

The non-empty subsets of A={27, 42, 30, 94} are {27}, {42}, {30}, {94}, {27,42}, {27,30}, {27,94}, …, {27,42,94}, {42,30,94}, {27,42,30,94}. The LCMs (least common multiples) of all these subsets are ...
2
votes
1answer
33 views

$x$ positive, rational but not an integer. $x^x$ irrational.

Let $x$ be positive, rational, but not an integer. That means $x$ can be written as $\frac{p}{q}$ with $p,q$ coprime, $p,q \neq 0$ and $q \neq 1$. Is $x^x$ always irrational? I think that this has to ...
0
votes
3answers
32 views

Can $p^{\frac{p}{q}}$ and $q^{\frac{p}{q}}$ both be rational with $p,q$ relatively prime and $p,q \neq 0$ and $p,q \neq 1$

Can $p^{\frac{p}{q}}$ and $q^{\frac{p}{q}}$ both be rational with $p,q$ as integers relatively prime and $p,q \neq 0$ and $p,q \neq 1$? I think so, but I am not able to prove it...
3
votes
0answers
67 views

How prime numbers are related to special functions?

We know that the Riemann zeta function is defined as $$\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s},$$ for all $\Re(s)>1$. Because of Euler product formula we also know that $$\zeta(s) = ...
2
votes
1answer
39 views

Number of integers of the form $3k+1$ in range $[a,b]$ [on hold]

How do I find the number of integers in the range $[a, b]$ that are of the form $3k+1$, where: $a,b,k$ are natural numbers. $a \le b$
2
votes
3answers
58 views

$\dfrac1a+\dfrac1b=\dfrac1c$, $a, b, c \in \mathbb{N}$ with no common factor, find all solutions [duplicate]

Given $\dfrac1a+\dfrac1b=\dfrac1c$, where $a, b, c \in \mathbb{N}$ with no common factor, find all solutions. Actually, you can think this question as a follow up of this one. Today, I saw this ...
2
votes
1answer
24 views

Solving a difference of powers equation

Is there a general method for solving this equation: $p^q - q^p = N$ Here $N$ is a positive integer.
0
votes
2answers
25 views

Is it true that $\sum\limits_{n=1}^N e^{in} = \frac{e^i(1-e^{iN})}{1-e^i}$ even if $| e^i | > 1$?

Is it true that $\sum\limits_{n=1}^N e^{in} = \frac{e^i(1-e^{iN})}{1-e^i}$ even if $| e^i | > 1$? I know this question is quite trivial and I will understand if it gets removed. I am trying to ...
4
votes
1answer
61 views

How many integers could be in such a way that any digits is not bigger than the left digits?

How many 4-digits integers could be in such a way that any digits is not bigger than it's left digits? I Try it with simulation, i get 714. anyone could describe a formula for me? My try:
0
votes
1answer
33 views

Is $\sum\limits_{\text{p prime}, p \geq 2}\frac{(-1)^{\frac{p^2-1}{8}}}{p}$ convergent or divergent??

Is $\sum\limits_{\text{p prime}, p \geq 2}\frac{(-1)^{\frac{p^2-1}{8}}}{p}$ convergent or divergent? So far I have that \begin{align} \sum\limits_{\text{p prime}, p \geq 2} ...
2
votes
1answer
34 views

The congruence has a solution

Sentence: If $a \in \mathbb{Z}$, then the congruence $x^2=a \pmod p, \forall p \in P$ has a solution $\Leftrightarrow$ $a=\square$ in $\mathbb{Z}$. If $a=\square$, then $\exists d \in \mathbb{Z}$ ...
4
votes
2answers
47 views

$(n+1)^{\textrm{st}}$ prime less than $2^{2^n}$

Using elementary means, show that the $(n+1)^{\textrm{st}}$ prime is less than $2^{2^n}$ please do not use fancier stuff like the prime number theorem or beyond. using this how can you show that ...
0
votes
0answers
26 views

The sets are equal

I want to show that $Z_p^*= \{ x \in Z_p | |x|_p=1 \}$. I have tried the following: Let $x \in Z_p^*$, then $x=a_0+a_1p+a_2p^2+ \dots \ \ \ \ \ \ \ \ 1 \leq a_i \leq p-1$. So, $x \in Z_p $. When ...
4
votes
0answers
33 views

$\sum_{a^2<p\leq (a+1)^2}p$ Summation of primes

$$\sum_{a^2<p\leq (a+1)^2}p$$ where p is prime. Are there some known bounds for this sum?
0
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0answers
28 views

Identity of the $p-$norm

A $p-$ norm of $\mathbb{Q}_p$ is a function $||_p: \mathbb{Q}_p \to \mathbb{R}$ $$x \neq 0, x=p^{w_p(x)}u \mapsto p^{-w_p(x)}$$ $$\text{ For } x=0 \Leftrightarrow w_p(x)=\infty \\ p^{-\infty}:=0$$ ...
1
vote
1answer
47 views

Which number appears directly below 2012 in a triangular number arrangement [on hold]

All of the positive integers are written in a triangular pattern, beginning with the following four lines and continuing in the same way: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Which number ...
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votes
0answers
19 views

Example of application of extended euclidian algorithm

Neymer has some (say, n) marbles (small glass balls) and he has going to buy some boxes to store them. The boxes are of two types: Type 1: each box costs c1 Taka and can hold exactly n1 marbles Type ...
8
votes
0answers
192 views

Murder at Hilbert's Hotel!

I'm sorry if this is a duplicate in any way. I doubt it's an original question. Due to my ignorance, it's difficult for me to search for appropriate things. This question is inspired by Exercise ...
3
votes
1answer
66 views

Convergence of infinite product of prime reciprocals?

Where pn is the nth prime number, does the infinite product $$\prod_{n=1}^{\infty}\left(1-\frac{1}{p_n}\right)$$ converge to a nonzero value? (Any help would be much appreciated!)
1
vote
1answer
82 views

Asymptotic estimate of sum of reciprocals of $\log(p_k)$

Fix $n$. The sum $S(n)=\sum_{p_k\leq n} \frac{1}{\log p_k}$ taken over primes less than or equal to n, clearly diverges as $n$ goes to infinity, being lower bounded by the sum of the reciprocals of ...
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votes
0answers
34 views

A problem related to least common multiple [closed]

The non-empty subsets of A={27,42,30,94} are {27}, {42}, {30}, {94}, {27,42}, {27,30}, {27,94}, …, {27,42,94}, {42,30,94}, {27,42,30,94}. The LCMs (least common multiples) of all these subsets are ...
0
votes
1answer
30 views

Decryption of a RSA encrypted message is not working.

Using RSA with e=13 (encrypting power), d=17 (decrypting power) & n=33 (RSA modulus) I noticed that once I decrypted the encrypted message it would be different then the original message. Why is ...
2
votes
0answers
34 views

Find nearest integer with certain prime factors

Trying to answer this question, I have sketched a strategy: Find an integer $N$ "near" to $10^9$ with factors $5$ and $7$, that is, $5^n\cdot7^m$ Conpute $k=N-\varphi(N)$ ($\varphi$ is the totient ...
5
votes
0answers
38 views

Find all positive integers $n$,$m$ with $n! = \frac{m(m-1)}{2}$ [duplicate]

Let $n$ and $m$ be positive integers. Find all solutions of the equation $n! = \frac{m(m-1)}{2}$. I found the following solutions: $(1,2)$, $(3,4)$, $(5,16)$, and tried to find more solutions with a ...
0
votes
0answers
25 views

The representations of numbers by decimals

I'm looking for books that talk about the representation of the integers by decimals, more specifically for prime numbers. I can't found anything yet, I read something in "AN INTRODUCTION TO THE ...
0
votes
0answers
26 views

Is there a natural interpretation of automorphic forms in terms of fractal geometry?

Disclaimer : This question is rather vague and thus might not be suitable for MathOverflow, so I prefer to ask it here. According to Wikipedia, an automorphic form is, roughly speaking, a ...
0
votes
1answer
19 views

Quotient of two rational sequences and the nature of its limit

Suppose we have two sequences of rational numbers, $(p_i)_{i=1}^\infty$ and $(q_i)_{i=1}^\infty$, and suppose $$\lim_{i\to\infty}\frac{p_i}{q_i}=c<\infty,$$ where $c$ is known. Are there any ...
1
vote
0answers
17 views

gcd of product of exponents of prime factors and product of prime factors

Let $n = \prod\limits_i p_i^{k_i}$. I want to express $$ \gcd(\prod\limits_i k_i, \prod\limits_i p_i) $$ as an arithmetic function (i.e get rid of gcd). Is that possible? Thanks!
1
vote
0answers
15 views

Prove identities-p norm

A $p-$ norm of $\mathbb{Q}_p$ is a function $||_p: \mathbb{Q}_p \to \mathbb{R}$ $$x \neq 0, x=p^{w_p(x)}u \mapsto p^{-w_p(x)}$$ $$\text{ For } x=0 \Leftrightarrow w_p(x)=\infty \\ p^{-\infty}:=0$$ ...
0
votes
0answers
16 views

Finding discriminant of this quadratic form

My question is what are the discriminants of $X^2+Y^2 $ and $X^2-Y^2$ over $\mathbb{R}$ and $\mathbb{C}$ and why? It should be $1$ and $-1$ respectively over $\mathbb{R}$. But shouldn't they be same ...
0
votes
2answers
40 views

An LCM related problem

If $A=\{27,42,30,94\}$ is a set, then all possible subsets from the set will be $\{27\}$, $\{42\}$, $\{30\}$, $\{94\}$, $\{27,42\}$, $\{27,30\}$, $\{27,94\}$, $\{42,30\}$, $\{42,94\}$, $\{30,94\}$, ...
-1
votes
1answer
18 views

Solution of congruence relation

The congruence relation $x^{n}≡2\pmod{13}$ has a solution for (a) n=5 (b) n=6 (c) n=7 (d) n=8 ? Are there any method to find the value of n?Putting the values of n & finding the solution is very ...
1
vote
4answers
105 views

Power series in $\mathbb{Q}_5$

Could you help me to find the first five positions of the power series in $\mathbb{Q}_5$ of $\frac{1}{2}$? How can I do this? Is there a general formula?