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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2
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0answers
26 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 cubes. The equation is $$x^3+y^3+z^3=\sigma^3$$ where $\sigma$ is a perfect number. Also either $x$ ...
4
votes
0answers
26 views

Which prime gaps are known to exist

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
votes
2answers
55 views

Finding the Probable Least Common Multiples of 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 ...
0
votes
0answers
7 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 ...
0
votes
2answers
23 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
36 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
votes
0answers
20 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
votes
0answers
21 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
votes
1answer
234 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 ...
1
vote
1answer
22 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
votes
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 ...
7
votes
0answers
67 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
votes
0answers
59 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
votes
0answers
46 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
votes
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
votes
0answers
58 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
40 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
votes
2answers
31 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
41 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
vote
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
votes
4answers
63 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
votes
1answer
100 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
votes
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 ...
-1
votes
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
votes
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 ...
6
votes
0answers
121 views
+50

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
vote
1answer
36 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
vote
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
vote
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
votes
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
37 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
301 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
29 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
19 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
115 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 ...
0
votes
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
votes
1answer
30 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 ...
-1
votes
0answers
34 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 ...
1
vote
3answers
105 views

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

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
0
votes
1answer
38 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
votes
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.
0
votes
3answers
17 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 ...
-1
votes
1answer
23 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 ...
-4
votes
0answers
11 views

Questions on tournament [on hold]

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
vote
1answer
25 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 : ...
1
vote
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
votes
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 ...