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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15 views

Strict total ordering

I'm not able to understand how the below relation is example of "strict total order". Consider a set $X = 2^Y$ where $Y = \{1,2,3,4,5,6,7,8,9\}$. The expected order of $X$ is for all $x, y$ ...
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2answers
24 views

Why $(10^ab+c)^{4d+1}-c \mid 10$?

I came across the following equation: $$x=(10^ab+c)^{4d+1}-c$$ Why is $x$ a multiple of $10$ for any natural number values for $a$, $b$, $c$ and $d$? The only progress I made was that $a$ could be ...
3
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0answers
29 views

Are there infinitely many $N^3$ (especially for prime $N$) that cannot be expressed as a sum of three positive cubes?"

Well few days ago i asked a question on perfect numbers and Tito Piezas III answered the question in a very intriguing way which has helped me to get a lead on it.But his answer and perfect numbers ...
3
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1answer
33 views

Two question about ${\mathbb Q}(\alpha)$ for $\alpha$ := $\sqrt[n]{7}$ + $\sqrt[n+3]{7}$

Let $\alpha$ = $\sqrt[n]{7}$ + $\sqrt[n+3]{7}$ with $n$ not divisible by $3$. Prove that $[{\mathbb Q}(\alpha) : {\mathbb Q}] = n(n + 3)$. Conclude that $\alpha$ is constructible if and only if $n = ...
2
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0answers
46 views

Trying to understand a step in Ramanujan's Proof of Bertrand's Postulate regarding the gamma function

My question relates to this step in the proof here: But it is easy to see that $$\log \Gamma(x)-2\log\Gamma(\frac12x+\frac12) \le \log\left\lfloor ...
2
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1answer
58 views

Multiples of 3 and 5. [on hold]

If we have the Tartaglia(Pascal) triangle in every row which numers are multiples of 3 which are even and which are multiples of 5?
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1answer
18 views

Understanding Bézout's identity

I'm trying to understand a proof of Bézout's identity ($gcd(a,b)=$ smallest linear combination of $a$ and $b$), and I'm having some trouble following the last step. The proof goes by: Let $m=sa+tb$ ...
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2answers
32 views

Is there a subset of natural numbers with a special property

Let set $A$ be an infinite big subset of the set $\mathbb{N}$ (set of natural numbers),it is not equal to $\mathbb{N}$ and it has the following property: For every $a$ that is not from the set $A$ ...
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1answer
17 views

How to solve second order congruence equation if modulo is not a prime number

the equation is $x^2 = 57 \pmod{64}$ I know how to solve equations like (*) $ax^2 +bx +c = 0 \pmod{p}$, where $p$ is prime and i know all the definitions for like Legendre's Symbol and all of the ...
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1answer
11 views

Partition of fractional parts where each sum of them has to be at least 1

Let $ a_1,\ldots,a_t \in \mathbb{Q} \setminus \mathbb{Z} $ be with $ \sum_{i=1}^t \lbrace a_i \rbrace \in \left[k,k+1\right) $ for some $ k \in \mathbb{N} $ with $ k \ge 4 $. Here $ \lbrace x \rbrace ...
0
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1answer
135 views

Selfy number couldn't exist?

For a positive integer $x$, let $S(x)$ denote the sum of the digits of $x$, and $l(x)$ denote the number of digits of $x$ (in base $10$). Now given a positive integer $n \ge 2$, it seems that there ...
3
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1answer
32 views

Integer solutions to a two variable equation.

For $m, n \in \mathbb{Z}$, show the only integer solutions to $f(m,n) = \displaystyle \frac{3^m(2^n+1)-2^{m+n}}{2^{m+n}-3^{m+1}}$ are $f(1, 2) = -7$, $f(0, 1) = -1$, and $f(0, 2) = 1$. More ...
5
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1answer
133 views

Can the cube of every perfect number be written as the sum of three cubes?

I found an amazing conjecture: the cube of every perfect number can be written as the sum of three positive cubes. The equation is $$x^3+y^3+z^3=\sigma^3$$ where $\sigma$ is a perfectnumber (well it ...
5
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0answers
59 views

Which prime gaps are known to exist [duplicate]

It is easily proved that prime gaps can be arbitrarily large by constructing the sequence of composites $(n+1)! + 2, (n+1)! + 3, \dots, (n+1)! + (n+1)$, which are divisible by $2, \dots, n+1$ ...
5
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3answers
89 views

Finding the possible Least Common Multiples of of numbers with Highest Common Factor 8

The Highest Common Factor of two numbers is 8. Which one of the following can never be their Least Common Multiple? The choices are as follow: A. 8 B. 12 C. 60 D. 72 The answer key states ...
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0answers
14 views

Asymptotics of $\sum_{\mathfrak{a}}\frac{n^{k-\epsilon}}{\mathfrak{N}\left(\mathfrak{a}\right)^{r\left(k-\epsilon\right)}}$

In this paper by Brian D. Sittinger, the following claim is made: For an algebraic number field $K$ with norm $\mathfrak{N}$, let $\epsilon=\left[K:\mathbb{Q}\right]^{-1}$. Then, taking the sum over ...
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2answers
27 views

all the squares in the multiplicative group $\mathbb{Z}_n^*$. [on hold]

I just want to know what this statement means: "all the squares in the multiplicative group $\mathbb{Z}_n^*$."
4
votes
2answers
46 views

Diophantine equation not solvable in $\mathbb{Q}$, but in $\mathcal{O}_p$

I'm trying to think of an example of a diophantine equation which can be solved in $ \mathcal{O}_p$ (meaning it can be solved $\mod p^k$ for all $ k $) for all prime $ p $'s, but not in $\mathbb{Q}$ ...
3
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1answer
37 views

Finitely Many Extensions of Fixed Degree of a Local Field

How does one show that there are only finitely many degree $n$ extensions of a local field? I understand how this follows from class field theory in the Abelian case but don't understand how to do the ...
0
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0answers
23 views

Hensel's lemma in $n $ variables

I'm trying to find a proof for the following formulation of Hensel's lemma: $$\text{Let } f \in \mathbb{Z}[x_1, \dots, x_n], a = (a_1, \dots, a_n) \text{ be such that (with } p \text{ prime)}$$ $$ ...
13
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1answer
255 views

Numbers that are divisible by the number of primes smaller than them

Let $\pi(n)$ denote the number of primes less than or equal to $n$ (a.k.a the prime-counting function). For certain values of $n$, the value of $\frac{n}{\pi(n)}$ is integer. Here are the first few ...
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1answer
23 views

Chevalley's theorem proof

I'm trying to prove Chevalley's theorem stating that $$ \text{If } f \in \mathbb{Z}[x_1, \dots, x_n] \text{ is a form of degree } r < n \text{,}$$ $$ \text{then there exists a nonzero solution of ...
0
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0answers
28 views

>Find all pairs of positive integers $(m, n)$, so that $1 + x + x^2 +\ldots+x^m \mid 1 + x^n + x^{2n} +\ldots +x^{mn}$

Find all pairs of positive integers $(m, n)$, so that $1 + x + x^2 +\ldots+x^m \mid 1 + x^n + x^{2n} +\ldots +x^{mn}$ I have to find $(m, n)$ such that ...
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0answers
70 views

Integer solutions to $\prod\limits_{i=1}^{n}x_i=\sum\limits_{i=1}^{n}x_i^2$

Given integers $x_1,\dots,x_n>1$. Let's assume WLOG that ${x_1}\leq\ldots\leq{x_n}$. I want to prove that the only integer solutions to any equation of this type are: $x_{1,2,3 ...
5
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1answer
73 views

Paul Erdős showed a simple estimate for $\pi(x) \ge \frac{1}{2}\log_2 x$; is it possible to tweak his argument to improve the estimate?

Paul Erdős gave a simple argument to show that $\pi(x) \ge \dfrac{1}{2}\log_2 x$. Is it possible to tweak the argument and get a better estimate? I am wondering how good an estimate for $\pi(x)$ can ...
2
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0answers
48 views

Any formula for the exact number of primes below a given bound?

Reading The music of the primes, the author relates that Riemann had figured out a formula giving exact number of primes up to a certain bound with no errors. Does such formula really exist? If ...
0
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3answers
30 views

Which residue classes modulo $p$ have exactly one square root?

Let $p>2$ be a prime. Which residue classes modulo $p$ have exactly one square root? Explain. I am having trouble understand the question. What does it mean for a residue classes to have exactly ...
2
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0answers
61 views

Prove that for $n\ge 6$ there is always a solution

We have the eation $\frac{1}{a_1^2} + \frac{1}{a_2^2}+...+\frac{1}{a_n^2}=1$. Prove that the equation has for $n\ge 6$ always natural solutions. Any $\frac{1}{x^2}$ can be displayed as sum of 4 ...
2
votes
1answer
42 views

Solve the eqation $p^8-p^4=n^5-n$

Solve the equation for primes $p$ and natural numbers $n$ $p^8-p^4=n^5-n$. For $p=2$ we get $n=3$, bjt for the next 5 prime numbers we get irational numbers. I cant prove (if its true) that there are ...
2
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2answers
32 views

Determine if the polynomial $P(x)=x^2-x+2 \mod p$ factors for the primes $p=5,7,11$, and $101$.

Determine if the polynomial $P(x)=x^2-x+2 \mod p$ factors for the primes $p=5,7,11$, and $101$. If it does factor for a particular prime provide a factorization. If not explain why. How would I be ...
3
votes
1answer
43 views

What proportion of the positive integers satisfy this number-theoretic inequality?

Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$, and let the abundancy index of $x$ be defined as $$I(x) = \frac{\sigma(x)}{x}.$$ My question is this: What proportion of the ...
1
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0answers
31 views

Sum of two squares - Number of steps in Fermat descent

If a prime $p$ can be written as the sum of two squares, then one can construct this representation via Fermat descent if we know an $x$ such that $x^2 \equiv -1 \mod p$. Is there a possibility to say ...
2
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4answers
66 views

Proving that $x^5 = x \pmod{10}$ for every integer $x$. [duplicate]

Show that $x^5 = x \pmod{10}$ for every integer $x$. How can I approach this? Should I use induction? I am stuck trying to get it in terms of $x+1$. Some feedback would be appreciated.
1
vote
1answer
70 views

Find all natural sequences $a_n=a_{a_{n-1}}+a_{a_{n+1}}$

Find all natural sequences for which holds $a_n=a_{a_{n-1}}+a_{a_{a+1}}$ a) for all natural numbers $n\ge 2$ b) for all natural numbers $n\ge 3$ I tried to do something with the caracteristic ...
0
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1answer
102 views

Finding the largest factorial of only three digits

I am using the following Python code to compute the above, but no results up to 16000!: ...
0
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0answers
26 views

Prove by induction on the binomial coefficient n choose m …

Prove by induction on $n$ that the binomial coefficient $\begin{pmatrix}n\\m\end{pmatrix}$ is the number of subsets of $I_{n}$ having size equal to $m$. The solution is as follows: So far it can be ...
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1answer
33 views

Sums of Divisors Function [on hold]

Can someone give me a clue where to start with this question: Show that $\sigma$ ( n ) = O ( n log n ) Where $\sigma$(n) is the sums of divisors function.
0
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1answer
33 views

2013th powered prime factors

Let $N$ be a positive integer. Prove that there exists a positive integer $n$ such that $n^{2013}-n^{20}+n^{13}-2013$ has at least $N$ distinct prime factors. factorize the polynomial for some ...
8
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1answer
179 views

Identity in Number Theory Paper

In this paper by Jerry Hu, he defines the function $$f_{s,k,i}\left(u\right)=\prod_{p\mid u} \left(1-\frac{\sum_{m=i}^{k-1}{s \choose m}\left(p-1\right)^{k-1-m}}{\sum_{m=0}^{k-1}{s \choose ...
1
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1answer
37 views

Completely self-contained (and as elementary as possible) introduction to Teichmuller Theory

Can you recommend a completely self-contained and elementary (as much as it can be) introduction to Teichmuller Theory?
1
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1answer
25 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
votes
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
votes
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 ...
0
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1answer
20 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,\ ...
0
votes
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 ...
7
votes
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 #$1$: ...
0
votes
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 ...
0
votes
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 ...
2
votes
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 ...