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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1answer
112 views

Asymptotic density and sum of the reciprocals

Let A and B be two infinite proper-subsets of the set of positive integers. Let A(n) denote the number of those elements of the set A , which does not exceed n ; we use similar definition for B(n) . ...
3
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1answer
476 views

Convert an unsolved number theory problem into a shorter program?

This problem of Waring's is unsolved: For all $n \ge 2 $, $\lfloor (\frac 32 ) ^n \rfloor + {3^n} \bmod{2^n} < 2^n $. (Kubina and Wunderlich have tested this up to $471,600,000$.) This can be ...
1
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1answer
186 views

Pell equation of the special form

I have tried several times to solve the following equation and finally, I was failed to complete. Help me to find the solutions of Pell equation $y^2-2x^2 = p^m$, where $p$ is prime and $8|(p-1)$ or ...
5
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1answer
173 views

Quotient of ring of integers

Let $R=\mathcal{O}(K)$ be the ring of the integers of $K=\mathbb{Q}[\zeta_8]$, where $\zeta_8=e^{2\pi i/8}=\sqrt{2}/2(1+i)$ is a primitive eighth root of unity in $\mathbb{C}$. It can be shown that ...
3
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1answer
234 views

History of the study of rational points on the circle

What is the first known instance of a mathematician parameterizing rational points on the unit circle by the slopes of rational lines going through a rational point on the circle?
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1answer
112 views

Number triangle

Can someone guide me as to how to go about working out a formula for this number triangle: ...
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0answers
158 views

Matiyasevich polynomial proof

Can someone provide a proof, or a link to a proof, of why does the Matiyasevich polynomial always generate primes for the nonnegative results? Any help will be appreciated.
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5answers
338 views

How can I compute the sum of $ {m\over\gcd(m,n)}$?

$$ \sum_{m =1}^n {m\over\gcd(m,n)}$$ For example, for 1 it is $${1\over\gcd(1,1)} =1;$$ for 5 it is $${1\over \gcd(1,5)}+{2\over \gcd(2,5)}+{3\over \gcd(3,5)}+{4\over \gcd(4,5)}+{5\over ...
3
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1answer
86 views

Is the ring $(Z_p[[X]] \otimes Q_p)/(X-p)^r$ principal?

Consider the ring $\mathbb{Z}_p[[X]] \otimes_\mathbb{Z} \mathbb{Q}_p$ and the ideal generated by $(X-p)^r$ (for some integer $r$). Is the following true : for all integer $r$, the ring $$ ...
2
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4answers
421 views

Exercise on Fundamentals of Divisibility: Factorization Domains

Set $R=\mathbb{Z}[\sqrt{10}]$. Show that in $R$ every element $\alpha\not=0$ is a product of irreducible elements, but $R$ is not a unique factorization domain. I have shown that $R$ is not a unique ...
0
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1answer
58 views

The probability of sum of $k$ randomly chosen numbers from $\mathbb{Z}_p$ being zero

Let $\{x_i\}_{i \in [k]}$ be randomly chosen independently from the uniform distribution over $\mathbb{Z}_p$. What is the probability that $\Sigma_{i\in [k]} x_i = 0$?
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3answers
451 views

Binary quadratic forms over Z and class numbers of quadratic fields.

What is the relation between the classification of binary quadratic forms over $\mathbb Z$, and the problem of finding the class numbers of quadratic fields? What would be a nice reference for this?
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0answers
287 views

Asymptotic FLT $\implies$FLT using ABC Conjecture

Edit: I'm beginning to suspect I either misread the sources or perhaps something wasn't stated, but it's my guess now that there is no nontrivial way of showing the ABC conjecture implies Fermat's ...
2
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2answers
65 views

Argue that $\phi(m_1*m_2)$ is strictly less that $\phi(m_1)*\phi(m_2)$ if…

Let $m_1,m_2\ge1$ be such that $\gcd(m_1,m_2) \ne 1$. Argue that $\phi(m_1*m_2)$ is strictly less that $\phi(m_1)*\phi(m_2)$. What I have so far: By example (which I'm not sure if that's how he ...
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2answers
42 views

Find the orders, if..

Suppose that $ord_ma=6$. Find the orders $ord_m(a^2)$ and $ord_m(a^5)$. Explain why your answers are correct. What I know: For $ord_ma=6$ I believe this is saying $a^6 \equiv 1 (mod m)$. Assuming ...
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1answer
85 views

Explain why $a^k\equiv1 \pmod {m_{1}\cdot m_{2}}$ if…

Supppose that $\gcd(m_{1},m_{2})=1$ and that for some $a$ and $k\ge1$ we have that $a^k\equiv1 \pmod{m_{1}}$ and that $a^k\equiv1\pmod{m_{2}}$. Explain why $a^k\equiv1 \pmod{m_{1}\cdot m_{2}}$.
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1answer
53 views

Find the indicated orders:

Find the indicated orders below: $$(a) ord_{8}3$$ $$(b) ord_{11}3$$ $$(c)ord_{20}3$$ $$(d)ord_{20}19$$ $$(e)ord_{35}1$$ I'm not sure what this is asking for. If someone could either give me a hint, ...
0
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1answer
32 views

Simultaneously smooth

I came across a problem recently which can be reduced to finding numbers $m$ such that $m$ and either $5m+1$ or $5m-1$ are $\{2,3,5\}$-smooth, i.e., of the form $2^a3^b5^c$ for nonnegative integers ...
2
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2answers
91 views

Which ice cream shop will get the most customers?

Consider ice cream shops numbered 1 to 1000 and same number of boys. The first boy goes to 1,2,3,4...1000. The second boy goes to 2,4,6...1000. 3rd goes to 3,6...,and so on for each boy. Which shop ...
4
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2answers
65 views

optimal solution for rational PI approximation

Consider 2 integers such that $a/b \approx\pi$. Let $c=|\pi-(a/b)|$ As $a$ and $b$ grow $c \to 0$. Now consider $d=abc$. Do $a$ and $b$ exist such that $d$ is less than it would be for any other ...
2
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1answer
173 views

Generalization of Euler's $\gamma$ constant

Let us define $H(k, n) = \displaystyle\sum_{i = 1}^{n} \frac{(\log i)^{k}}{i}$. We want to show that $H(k, n) - \displaystyle\frac{(\log n)^{k + 1}}{k + 1}$ converges as $n \rightarrow \infty$. ...
4
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2answers
313 views

at least $a\sqrt{n}$ solutions

Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed. Consider a number $0<a<2\pi$. How can we show that there ...
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2answers
67 views

If $N(\alpha)$ = $p$ where $p$ is an odd prime, then is ($\alpha, \bar\alpha$) = $\mathbb{G}$?

Here, $N(\alpha)$ stands for the norm of $\alpha\in\mathbb{G}$, $\mathbb{G}$ is the set of Gaussian Integers, and ($\alpha, \bar\alpha$) is the ideal generated by $\alpha$ and $\bar\alpha$. In other ...
2
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3answers
265 views

primitive pythagorean triple, $\gcd(b,c)=1$ proof

Let $s, t \in \mathbb{Z}^+$, be both odd, $s > t \geq 1$, $\gcd(s, t) = 1$. If we set $$a = st,\quad b = \frac{1}{2}(s^2 - t^2),\quad c = \frac{1}{2}(s^2 + t^2)$$ then $(a, b, c)$ is primitive ...
0
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2answers
239 views

Divisibility by prime numbers

Suppose prime number $p$ and an arbitrary number $r$ is given. What methods do you know for detection that $r$ is divisible by $p$?
1
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1answer
53 views

What are the analogues of 'transfinitesimal' numbers?

There are such transfinite numbers as $\aleph$ and $\omega$. Do they have infinitesimal analogues? I'm especially interested in ...
0
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1answer
122 views

How to associate non-natural numbers with set features/relations?

Pythagoras said "everything is number". Therefore, it might seem that sets (everything) can be described/identified using numbers. But does it really make sense to associate non-natural numbers with ...
3
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2answers
377 views

Solving a mathematical recursion to find explicit function

I came across the recursive sequence $$ \begin{align} a_{n+1}&=(r-2)a_n+(r-1)b_n\;,\\ b_{n+1}&=a_n\;, \end{align} $$ and the explicit formula $$ a_n=(-1)^n(r-1)+(r-1)^n\;. $$ I saw that ...
2
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2answers
988 views

Expressing a Non Negative Integer as Sums of Two Squares

I'm writing a code in C that returns the number of times a non negative integer can be expressed as sums of perfect squares of two non negative integers. ...
7
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1answer
396 views

Undergraduate Research

I am currently writing a couple of undergraduate papers about primes and irrational numbers, and my advisor keeps saying that I need to motivate the topics and include a discussion at the end. Can ...
1
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1answer
55 views

Size of factors in number ring

Let $R$ be the ring ${\mathbb Z}[\sqrt{2}]$. For $z\in R$, $z=x+y\sqrt{2}\in R$ with $x$ and $y$ in $\mathbb Z$, put $\|z\|={\sf max}(|x|,|y|)$ and $$D_z=\left\{ (a,b) \in R^2 : ab=z,\ a \text{ and ...
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0answers
137 views

Has the $\Gamma$-like function $f_p(n) = 1^{\ln(1)^p} \cdot 2^{\ln(2)^p} \cdot \ldots \cdot n^{\ln(n)^p} $ been discussed anywhere?

In an older fiddling with the gamma-function (expanding on the idea of sums of consecutive like-powers of logarithms, similarly as the bernoulli-polynomials for the sums of like powers of consecutive ...
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2answers
48 views

Evaluate $m(S)$.

Let $S$ be the set of all $n$-digit ($n > 1$) numbers $N$ such that: $N$ is formed from digits $1,2,4,5$ and the difference between any two consecutive digits of $N$ is 1. $N$ is divisible by 11. ...
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2answers
245 views

Is there a way to show that $\sqrt{p_{n}} < n$?

Is there a way to show that $\sqrt{p_{n}} < n$? In this article, I show that $f_{2}(x)=\frac{x}{ln(x)} - \sqrt{x}$ is ascending, for $\forall x\geq e^{2}$. As a result, $\forall n \geq 3$ ...
0
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1answer
52 views

Exhibit this correspondence.

We know that there is a one-to-one correspondence between the $r$ with $0\le r\lt20$ with $gcd(r,20)=1$ and the pairs $(r_1,r_2)$ where $0\le r_1\lt4$, where $0\le r_2\lt5$ such that $gcd(r_1,4)=1$ ...
3
votes
4answers
100 views

Deduce a unique number from number

I work on the tree where each node has N children. In my case each node has a unique identifier. i want to deduce an identifier of father node from the child identifier. So, we can add an ...
0
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2answers
181 views

given average score,find the selected numbers

Suppose average score of 46 scores selected from 1,2,3,4,5 is 1.65.Is there any way to find out which are the scores that are selected.
2
votes
2answers
302 views

What is the use of such concepts as potential infinity and actual infinity?

I'm aware of such mathematical concepts as and potential infinity and actual infinity. But I do not understand how those concepts are being used. Are there any applications to such concepts? Are there ...
2
votes
1answer
105 views

using totient function in proof

if $N=pq$ where $p$ and $q$ are different primes. Given $e$ and $d$ in $\{1,\ldots,\phi(N)-1\}$ for which holds that $ed=1\pmod {\phi(N)}$.Then we have $a^{ed}=a\pmod N$ for all non-invertible $a$ in ...
0
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1answer
76 views

Quadratic equation congruence

I am trying to find relation of pairs $(x,y)$ and $(x_1,y_1)$ such that $f(x,y)\equiv f(x_1,y_1)(\mbox{mod }n)$ where $f$ is a quadratic expression like $ax^2+bxy+cy^2+dx+ey+f$ Thanks in advance
5
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1answer
186 views

How to prove the equivalence between the two statements of ABC conjecture?

The ABC conjecture stated by wikipedia says the following statements are equivalent: I. For $\epsilon>0$, there are finite coprime triple $(a,b,c)$ satisfying $a+b=c$ such that ...
0
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1answer
103 views

Let $p \ge 5$ be a prime number. Find the largest length of an arithmetic progression satisfying the following

Let $p \ge 5$ be a prime number. Find the largest length of an arithmetic progression, of positive ratio, of positive integers whose terms do not contain the digit $1$ in their p-adic expansion.
2
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1answer
263 views

Assume there exists infinitely many prime numbers $l$ such that $2^l -1$ is NOT a prime, show the following

Assume there exists infinitely many prime numbers $l$ such that $2^l-1$ is NOT a prime, prove that ther exists infinitely many pairs $(p,q)$ of DISTINCT prime numbers $p \neq q$ s.t. $p\mid ...
3
votes
2answers
98 views

When does a binary quadratic form represent 1 or -1

Let $a,b,c$ be integers. Is there a reasonably concise condition on $(a,b,c)$ which ensures that $$ax^2+bxy+cy^2=\pm 1$$ has a solution in integers $x,y$? In addition to direct answers I would also ...
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2answers
274 views

Is there such math concept as potential zero?

It seems that I need to use concept of potential zero in my work and I want to know whether I could reference some other works in order to fully understand what I'm dealing with. Specifically for my ...
1
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1answer
185 views

Can you determine a formula for this problem?

Given: A list of integers is there.Now there are 2 buckets -bucket A and bucket B.This step is repeated as long as there are numbers left in the list.Integers from start or end of the list are ...
1
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1answer
147 views

Can one find 4024 positive integers such that the number “2013” cannot divide the sum of any 2013 of them

As stated on the title, Is it possible to find $4024$ positive integers such that the sum of any $2013$ of them is not divisible by $2013$? I used to assumed they have to be distinct. Being ...
4
votes
2answers
848 views

Trying to prove that $p$ prime divides $\binom{p-1}{k} + \binom{p-2}{k-1} + \cdots +\binom{p-k}{1} + 1$

So I'm trying to prove that for any natural number $1\leq k<p$, that $p$ prime divides: $$\binom{p-1}{k} + \binom{p-2}{k-1} + \cdots +\binom{p-k}{1} + 1$$ Writing these choice functions in ...
6
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0answers
77 views

Statement about Woodal primes.

A Woodal number is an integer of the form $n 2^{n}-1$. A Woodal prime is an integer that is both a prime and a Woodal number. Let $p$ be a prime of the form 1 mod 4. Then $p 2^{p} -1$ is never a ( ...
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votes
2answers
109 views

Does this proposition really prove that n is not a prime?

In my opinion, it doesn't make sense because I can't think of any number that meet that criteria for the number 4, which is not prime. a must be different from b, right? ...