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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1answer
26 views

Series involving primes

Trying to find an asymptotic bound for the series $$ S(x) =\sum_{p\leq x}\frac{\varphi(p-1)}{(p-1)p} $$ as $x \rightarrow \infty$. Of course $$ \frac{\varphi(p-1)}{p-1} =\prod_{q\mid ...
2
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1answer
24 views

sucessive primes with distance greater than k

I am studying bounds in prime gaps and I would like to gather as much information as I could. I am just an undergraduate student, it's not a very important project, I am just doing it by curiosity. I ...
1
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1answer
25 views

2013th powered sequence

Let $a_1$, $a_2$, ... be a sequence of integers defined recursively by $a_1=2013$ and for $n \ge 1$, $a_{n+1}$ is the sum of the 2013th power of the digits of $a_n$. Do there exist distinct positive ...
0
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1answer
18 views

Find $n$ with equalities of his divisors

Let for a natural number $n$ be $d_1<d_2<...<d_k$ his divisors, where $d_1=1,d_k=n$. Find all n so that $d_5-d_3=50$ and $11d_5+8d_7=3n$. From the second eqation I got that $d_{k-6}\le 6$ so ...
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1answer
21 views

Two recurrence relations gcd proof

Let $q_1, q_2,\ldots$ be a sequence of integers with $q_i\gt0$ for all $i\gt 1$. Define $(a_n)_{n\ge-1}$ and $(b_n)_{n\ge-1}$ by the following recurrence relations: $a_{-1}=0,\ a_0=1,\ b_{-1}=1,\ ...
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2answers
39 views

Solve $\frac{x}{m}=\lfloor{x^{2/3}}\rfloor+ \lfloor{x^{1/2}}\rfloor+1$

Prove that for every natural number $m$ there is a natural solution for the eqation $\frac{x}{m}=\lfloor{x^{2/3}}\rfloor+ \lfloor{x^{1/2}}\rfloor+1$ Beside the typical inequality I can't get nothing ...
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2answers
302 views

Consecutive numbers that share the same sum of prime factors

Let $f(n)$ denote the sum of the prime factors of $n$ (with multiplicity). I have been looking for pairs of consecutive numbers $n,n+1$ such that $f(n)=f(n+1)$. Case #$1$: ...
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2answers
37 views

Prove if $ord_p(d) < ord_p(n)$ then d divides n

I have to prove that $d$ divides $n$ if and only if $ord_p(d)\leq ord_p(n)$ I have already proved that $ord_p(d)\leq ord_p(n)$ if $d$ divides $n$ but I am struggling to prove the converse. Can ...
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0answers
20 views

Diophantine equation 3-rd degree.

When I decided this Diophantine equation, it became clear. If the coefficients are expressed as follows. $$b(x^3+y^3)=az^3$$ Where $$b=q^2+3n^2$$ $$a=2(q^2-3n^2)$$ When you can represent the ...
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2answers
118 views

Topology on $Z_p$

let $Z_p$ denote the $p$-adic integers, then it has a topology as a subspace of $\prod_nZ/p^nZ$, where $Z/p^nZ$ is given the discrete topology. (reference I posted before: Why $Z_p$ is closed.) Now ...
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1answer
51 views

Proof of Hensel's Lemma not clear

If you look at the following proof of Hensel's Lemma http://isites.harvard.edu/fs/docs/icb.topic1472247.files/Hensels%20lemma.pdf you will see that the author determines the conditions which these ...
0
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1answer
31 views

Summation of a finite series of unit fractions

Let's say I have a series of unit fractions, $\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}$, and we know that $a_{1} + a_{2} + \cdots + a_{n} = g$. Is there a general method or formula to ...
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0answers
37 views

Is there an efficient method to search prime factors near $9^{9^9}$?

Since, the number $9^{9^9}$ is very special, is there a better method to search prime factors for a number near $9^{9^9}$ than simply trial division ? Especially, I searched prime factors of ...
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3answers
105 views

can π be considered as a rational number without knowing its value [closed]

Can we write π as π\1 and consider it as rational without actually knowing the value of pi I think pi is just a symbol and its rationality actually depends upon its value
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1answer
40 views

On no. of solutions of product of positive integers equal to sum

$n \ge 2$ be an integer , let $a(n)$ be the no. of solutions in positive integers of $x_1+x_2+...+x_n=x_1x_2...x_n ; x_1 \le x_2 \le ... \le x_n$ , then is it true that $a(n+1)=1 \implies n$ is ...
2
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2answers
51 views

Is this divisibility problem correct?

Let $n$ be a natural number and let $1 \le a_1<a_2<...<(a_k=n)$ be all of its divisors. Find all $n$ such that $a_2^3+a_3^2-15=n$ . It seems impossible to find all such numbers.
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3answers
18 views

Greatest common divisor of an integer 'a' and it's sum with 2.

I need to prove that the $\gcd(a, a+2)$ equals either 1 or 2. Intuitively this makes sense to me. If a is an odd integer then the gcd is 1, if a is even, the gcd is 2. I'm having trouble writing a ...
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1answer
25 views

Number of submatrices of sum K

I have an array $A[]$ of N elements ($N<=1000$, $-1000<=A[i]<=1000$). We define a Matrix M such that $M[i,j]= A[i]*A[j]$. In the resulting matrix $M$, we have to count the number of ...
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0answers
11 views

Questions on tournament [closed]

In a cricket tournament,a total of 15 teams participated.Australia won the tournament by scoring the maximum number of points.The tournament is organised as single round robin one-where each team ...
1
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1answer
27 views

Efficient Method/Algorithm to Compute the P-adic Ordinal of a Positive Integer

I am currently writing my master's thesis at Cal Poly Pomona, and am currently investigating the ruler sequence for a prime base. The ruler sequence for base $2$ is : ...
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2answers
69 views

Why $Z_p$ is closed.

Let $A_n=\mathbb{Z}/p^n\mathbb{Z}$ be a ring and $p$ is prime, $\phi_n: A_n\rightarrow A_{n-1}$ be a natural homomorphism (Elements of $A_{n}$ define in an obvious way elements of $A_{n-1}$). Define ...
2
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1answer
41 views

the number of zero divisors in polynomial ring

I was looking for an answer on the question How much zero divisors are in the ring $\dfrac{\mathbb{Z}_3[x]}{(x^4 + 2)}$? when I came up with the brilliant/hack-isch idea that it might just be ...
2
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1answer
77 views

Prove that there are infinitely many real numbers.

Here it goes: Assume that there is an upper bound first for the set $\mathbb{R}$ let $\alpha = \sup \mathbb{R}$ So assume $\alpha = \sup \mathbb{R}$ Therefore $\alpha \ge x$ for $x \in \mathbb{R}$ ...
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2answers
23 views

Proof of well ordering principle for the set of positive integers with directly using the principle of induction and not strong induction

Can we prove well ordering principle for the set of natural numbers (positive integers ) with directly using the principle of induction i.e. $( S \subseteq \mathbb N ,1 \in S \space \&\ n \in S ...
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0answers
34 views

Orbits of left-multiplication from $\mathrm{PSL}_2(\mathbb Z)$ on $\mathbb Z^{2\times 2}$

I am trying to learn moduli space of elliptic curves from different resources. But only a spacial class of elliptic curves where the lattice in the plane has "integer vectors" as generators. To study ...
6
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0answers
67 views

Applying iterated function on the sum of the squares of the prime factors of $30$

Let $f(n)$ denote the sum of the squares of the prime factors of $n$ with multiplicity. For example, $f(60)=f(2\cdot2\cdot3\cdot5)=2^2+2^2+3^2+5^2=42$. Denote the iterated function ...
1
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1answer
36 views

Let $x=0,a_1a_2a_3\cdots a_i\cdots$ be a number such that $a_1=0$, $a_i=1$ if $i\in\mathbb{N}$ is a prime number and $a_i=0$ otherwise.

Let $x=0,a_1a_2a_3\cdots a_i\cdots$ be a number such that $a_1=0$, $a_i=1$ if $i\in\mathbb{N}$ is a prime number and $a_i=0$ otherwise. So $x=0,01101010001\cdots$. Is $x$ a rational number? How can I ...
1
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1answer
17 views

Showing $\sum_{j \geq 2}\sum_{x^{1/j} \leq p \leq x}\frac{1}{jp^{j}} = O\left(\frac{1}{\log x}\right)$

Let $p$ denote a prime. Suppose I am given the asymptotic that $$\sum_{1 \leq n \leq x} \frac{\Lambda(n)}{n\log n} = \log\log x + \gamma + O\left(\frac{1}{\log x}\right),$$ why is $$\sum_{2 \leq p ...
7
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1answer
60 views

Reason for LCM of all numbers from 1 .. n equals roughly $e^n$

I computed the LCM for all natural numbers from 1 up to a limit $n$ and plotted the result over $n$. Due to the fast-raising numbers, I plotted the logarithm of the result and was surprised to find a ...
2
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0answers
63 views

The number of solution of a Diophantine equation

If we fixe $n\in \mathbb{N}$. I was wondring if there is an estimation of the number of the integer solutions of the equation : $$x_1^2+x_2^2+\cdots+x_n^2=n^3 $$ where $x_i>0$ for all ...
2
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0answers
79 views

$f(x)\sim 1/x \implies (1+f(x))^x\to e$, but what family of functions maximizes the speed of convergence from below?

This problem is subordinate to finding out if $$\left(1+\frac{\log p_{n+1}}{p_n}\right)^{p_{n+1}/\log p_n},$$where $p_n$ is the $n$-th prime, never stabilizes above or below its limiting value, which ...
1
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1answer
40 views

Carmichael number satisfying three conditions

Is there a composite number $n$ that satisfies these conditions? $2^{1023} \le n < 2^{1024}$, i.e. $n$ is a $1024$-bit number $n$ is not divisible by the first $100,000$ primes. i.e. $n$ is ...
0
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1answer
72 views

Integers Positioned Around a Circle [duplicate]

Nine distinct positive integers are arranged in a circle such that the product of any two non-adjacent numbers in the circle is a multiple of n and the product of any two adjacent numbers in the ...
0
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1answer
46 views

Constructing idele from a rational number.

I am a novice to concept of idele ,despite the fact that I have gone through all its expositions in standard literature. Excusing my ignorance,suppose I take q=396000. Does it mean that the idele q= ...
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0answers
33 views

Definition of Natural Numbers with one statement.

Is there a way that we can use mathematical or logical axioms to define natural numbers with one statement? In planar geometry there are several basic notions which can define everything with a finite ...
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0answers
21 views

On the Brent improvement over the Pollard rho factoring method

I try to understand the Brent improvement over the Pollard rho factoring method. see (R. Brent, "an improved monte carlo factorization algorithm", BIT, (20), pp. 176-184, 1980) available at the ...
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0answers
112 views
+50

How to calculate this sum like Gauss sum.

I would like to calculate the following sum, which looks like a Gauss sum. Let $n$ be a natural number and let $a,b$ be integers. Denote by $e(x)=e^{2\pi i x/n}$. Consider the sum $$ \sum_{1 \leq j, ...
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0answers
60 views

Solve $x^2 + 2 = y^3$ for integer $x$ and $y$ [duplicate]

I am asked to find all integers $x$ and $y$ which satisfy $x^2 + 2 = y^3$. I am given the hint that I should work in the unique factorization ring $\mathbb{Z}[\sqrt{-2}]$. So I could write the ...
3
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3answers
85 views

find a number such that, for all $a$ in $\{0,…,1926\}$, $a^x \equiv a \mod 1926$.

I don't want the answer, but I need some help on how to figure out the answer. If you could point me in the direction of a useful math theorem or technique it would much much appreciated. Also, I am ...
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2answers
32 views

On the Diophantine Equation $(x-h)^2+(y-k)^2=c$

I am just curious about the equation of the circle centered at (h,k) whose form is we know $(x-h)^2+(y-k)^2=r^2$. If we consider its solution over the set of integers then we have a Diophantine ...
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3answers
95 views

Prove that $2$, $3$, $1+ \sqrt{-5}$, and $1-\sqrt{-5}$ are irreducible in $\mathbb{Z}[\sqrt{-5}]$.

So the Norm for an element $\alpha = a + b\sqrt{-5}$ in $\mathbb{Z}[\sqrt{-5}]$ is defined as $N(\alpha) = a^2 + 5b^2$ and so i argue by contradiction assume there exists $\alpha$ such that $N(\alpha) ...
2
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1answer
32 views

Embeddings of a subfield of $ \mathbb{C} $

I'm trying to understand / solve the following problem: Let $ L \subset \mathbb{C} $ be a field and $ L \subset L_1 $ its finite extension ($ [L_1 : L] = m $). Prove that there are exactly $ m $ ...
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2answers
22 views

How to prove that N(u) = 1 if and only if u is a unit in $Z[\sqrt-5]$

The norm of an element $u=a+b\sqrt-5$ in $Z[\sqrt-5]$ is defined as $N(u)= a^2 +5b^2$, now if $N(u) = 1$ then $a^2+5b^2 = 1$ but then how would i prove that it's a unit !?
2
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1answer
98 views

Why don't we use Presburger's arithmetic instead of Peano's arithmetic?

I was reading about quantifier elimination and discovered the Presburger Arithmetic, the article mentions two points about it: It is decidable, complete and consistent. It omits multiplication ...
2
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0answers
81 views

The curse of Collatz [closed]

With this playful title I wanted to open a discussion on a personal attempt to prove this conjecture after several sheets and energy wasted (I'm not English, sorry if my language isn't perfect). The ...
1
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2answers
57 views

Which arrangement produces the largest number?

I learnt that the power tower $2\uparrow3\uparrow4\uparrow...\uparrow n$ is larger than any power tower with a different order of the numbers $2,3,4,...,n$. Is this also true for conway-chains and ...
3
votes
1answer
79 views

Solve the eqation $a^3+2b^3+4c^3=6abc+1$

Find all integers $a,b,c>2010$ so that $a^3+2b^3+4c^3=6abc+1$. If there are no solutions then prove it. As for now I only tried to use the identity ...
1
vote
1answer
65 views

How to evalute the $\sum_{n=1}^\infty \dfrac{\phi(n)}{2^n -1}$?

In the above question $\phi$ is Euler's phi-function. This problem belongs to IMO shortlist. All my efforts doesn't lead to any good result.
2
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4answers
64 views

Can one generate a sequence of natural numbers whose density has a given distribution?

Suppose $\{ p_{k} \}$ is a collection of real numbers with the following properties: 1) $p_k \in (0,1)$ $~~~~$(i.e. $0$ and $1$ are not allowed values) 2) $\sum_{k=1}^{\infty} p_k =1$ An ...
0
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1answer
32 views

GCD : Number Theory Problem

I was working my way through some number theoretic proofs and being a newbie am stuck on this problem : If (x, 4) = 2 and (y, 4) =2, then (x + y, 4) = 4 where (a,b) denotes gcd of a & b ...