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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0
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0answers
18 views

Is it a case of a mixed-integer problem?

I have been faced with a problem for quiet some time and don't know how to go further. The problem is as follows and would appreciate any help on formatting this problem in terms of mixed-integer ...
0
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0answers
19 views

number solution $x^2+y^2 \leq n^2$ [on hold]

As we know number solution positive integer of solution the inequality $x^2+y^2 \leq n^2$ $n$ is integer number
1
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0answers
30 views

Question about series and how the pattern idea works

Two Questions: When you are given: $1, 2, 3, .... , n$ How do you know that in the $...$ that it continues the $x_{n-1} + 1$ pattern? Is it the definition of series? Secondly: Do partial sums ...
-1
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0answers
48 views

Is the polynomial $X^{32} + 1$ irreducible? [on hold]

I think this question is interesting for Fermat number. Is this polynomial irreductible in $Z[X]$ ?
2
votes
1answer
44 views

Fermat solved $x^2+2=y^3$ by infinite descent?

In a letter to Christiaan Huygens entitled "on problems in the theory of numbers: a letter to Christiaan Huygens", Fermat claism that he solved the diophantine $x^2+2=y^3$ using infinite descent. Here ...
0
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1answer
44 views

Is it possible to find $n-1$ consecutive composite integers

Given an integer $n\geq 2$ ,can we always find an integer $m$ such that each of the $n-1$ consecutive integers $m+2,m+3,.....,m+n$ are composite?
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0answers
28 views

Count of 1's in the binary notation of a number

Let $f(x)$ be the count of 1's in the binary notation of number $x$ . Find minimal $g(x)$ for what $f(3^n-1)\le g(n)$ for all $n\ge1$ . I think that $g(x)=x$ but I can't prove it . For $n=1,\ldots,10$ ...
2
votes
0answers
42 views

Mellin transform on $\mathbb{Z}[\omega]$

Let $\omega=\frac{-1+i\sqrt{3}}{2}$ be a complex cube root of unity. The Eisenstein integers $\mathbb{Z}[\omega]$ (a unique factorization domain) are of the forms $a+b\omega$ where $a$ and $b$ are ...
3
votes
1answer
35 views

Calculating zeta functions over a field

I am learning about zeta functions and have been trying the following example: Calculate the zata function of $x_0x_1-x_2x_3$ over $\mathbb{F}_p$. Does there exist an easy formula for calculating ...
1
vote
0answers
16 views

Inertia Degree in Cyclotomic Extensions

Let $\zeta$ be a primitive $l$th root of unity, where $l$ is prime. If $p$ is another prime number, let $f$ be the order of $p$ in $U(\mathbb{Z}/l \mathbb{Z})$. Then in $\mathbb{Z}[\zeta]$, $p$ ...
1
vote
2answers
29 views

The set $\mathbb{Z}$ is totally ordered

Having the following definition of the $\leq$-Relation in $\mathbb{Z}$: For $a, b\in \mathbb{Z}$ we define $$ a \leq b : \iff b-a \in \mathbb{N} $$ Show that $(\mathbb{Z}, \leq)$ is totally ...
8
votes
2answers
169 views

On the difference between consecutive primes

Let $(p_n)$ be the sequence of prime numbers and $g_n = p_{n+1} - p_n$ Question: Is it known that $g_n \le n$? Remark: it's known that $g_n < p_n^{\theta}$ with $\theta = 0.525$ for $n$ ...
0
votes
2answers
63 views

Set of numbers which can not be represent as $a_1^n+a_2^n+…a_n^n$

Consider the set of natural numbers which can not be represented as $a_1^n+a_2^n+....a_n^n$ where $a_1,...,a_n$ are non-negative integers , n - given natural number . Is this set infinite or not?
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votes
2answers
46 views

Beautiful little number theory prob

Solving (a,b) + [a,b] = ab for natural a,b. How many possible a's are there. I only know that (a,b)[a,b] = ab. Tried factoring out (a,b), but can't derive from it.
0
votes
1answer
18 views

Does there exist a Unit Matrix for a m x n matrix?

By definition, a Unit/Identity matrix (I) is a matrix such that, I A = A I = A If the matrix A is of dimension m x n, then unit matrix in IA must be of dimention m x m, while in A I should be of ...
3
votes
0answers
50 views

Proving that table totals can always be preserved with ceiling and floor

$\begin{array}{|c|c|c|c|} \hline 11.998& 9.083 & 2.919 & &24 \\ \hline 12.983&10.872&3.145&&27\\ \hline 1.019&2.045&0.936&&4\\ \hline & & ...
5
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4answers
92 views

how do I find the continued fraction of root n ?? [duplicate]

I saw a site where they explained it.but it required calculator.I want to do it without calculator. Can anyone please help me?
3
votes
0answers
40 views

Lucas' Theorem for $p$-adic integers?

Does Lucas' Theorem hold for $p$-adic integers? More specifically, does it specifically hold for the case that, given a $p$-adic integer: $$x = x_0 + x_1p + x_2p^2 + \cdots = \sum_{i=0}^\infty ...
1
vote
1answer
26 views

Something similar to Euler's theorem

If $p$, $q$ are not equal primes. $n=pq$, $\varphi(n) = (p − 1)(q − 1)$, $d = \gcd(p − 1, q − 1)$. Is it true that for any $a$ such that $\gcd(a, n) = 1$ holds $a^{\frac{\varphi(n)}{d}} \equiv 1 ...
0
votes
1answer
11 views

Show that $\frac{1}{e^\gamma \text{log }x + O(1)} = \frac{1}{e^\gamma\text{log }x} + O\left(\frac{1}{(\text{log }x)^2}\right)$

Show that $\frac{1}{e^\gamma \text{log }x + O(1)} = \frac{1}{e^\gamma\text{log }x} + O\left(\frac{1}{(\text{log }x)^2}\right)$ I'm using one of Merten's estimates in a proof, the one that states ...
1
vote
1answer
26 views

Why is it impossible that $\frac{\phi(n^*)}{n^*} < \frac{\phi(n)}{n}$ when $n^* < n$

Why is it impossible that $\frac{\phi(n^*)}{n^*} < \frac{\phi(n)}{n}$ when $n^* < n$ and $n$ has $k$ prime factors, and $n^*$ is the product of the first $k$ prime factors? I understand that ...
-1
votes
0answers
12 views

Square classes of transcendental extension of p-adic fields.

Let $k = \mathbb{Q}_p(t)$ with $p \neq 2$. What is known about the order of $k^*/k^{*2}$ ? In the case $k = \mathbb{Q}_p$ we have that $k^*/k^{*2}$ is isomorphic to the Klein four group. So i guess ...
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0answers
29 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
45 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
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1answer
31 views

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

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
45 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 ...
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votes
0answers
22 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
51 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
31 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
54 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
66 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
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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
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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
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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...
2
votes
0answers
65 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
37 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
60 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 $\exists d \in \mathbb{Z}$ ...