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
votes
2answers
40 views

the solution to $x^2=49+k\cdot12288$

I have computed $x^2=49+k\cdot12288$ for $k=0$ to an arbitrarily large integer and found $x$ has an integer solution only for $k=0$. Can someone proove that for $x^2=49+k\cdot12288$, $x$ has only ...
-1
votes
0answers
74 views

Why do we care about representaion of primes?

Im currently trying to figure out the genesis of quadratic reprocity by using Cox and Lemmermeyers books. I also got a copy of some works of Fermat but it is in German. It seems like there is some ...
0
votes
2answers
17 views

Congruence with binomial

I tried to prove this by induction on $k$. But I did not manage Let $p$ be a prime. For every $k\in\{0,\cdots,p-1\}$, one has $\binom{p-1}k\equiv(-1)^k\pmod p$. By Wilson theorem, it suffices to prove ...
0
votes
0answers
10 views

Ring structure of tuples mod k

Consider a vector of n integers $$A= a_1, a_2, ... a_n$$ Such that for another vector $$B= b_1,b_2... b_n$$ $$AB^T \equiv 0 \mod k$$ For an integer k. I was playing around with these structures ...
1
vote
1answer
27 views

Question around the following relation: $T(n,1) = n$, for a positive integer $n$, and for all $k\geq 1,\ T(n,k+1)=n^{T(n,k)}$.

I'm beginning the studies on number theory and then i'm facing the following problem that i couldn't solve yet: given a positive integer $n$ and being $T(n,1)=n$ and, for all $k\ge1$, ...
0
votes
0answers
31 views

Evaluate the following sum using the hyperbola method

This is an exercise from Iwaniec and Kowalski's book Analytic Number Theory: Prove that $$\displaystyle \sum_{n \leq x} \tau(n^2 + 1) = \frac{3}{\pi}x \log x + O(x).$$ The constant $3/\pi$ is quite ...
4
votes
5answers
65 views

An integer when divided by $5$ and $13$ leaves residues $4$ and $7$ respectively

Find an integer when divided by $5$ and $13$ leaves residues $4$ and $7$ respectively. (Without Modular Arithmetic). I don't know if it is right, but i got this $$n=5x+4=13y+7$$ ...
0
votes
1answer
54 views

Why is this function an embedding?

We have the canonical function $\epsilon_p: \mathbb{Z} \to \mathbb{Z}_p, x \mapsto (\overline{x})_{k \in \mathbb{N}_0}=(\overline{x}, \overline{x}, \overline{x}, \dots )$. The function $\epsilon_p: ...
1
vote
2answers
36 views

Find $x$ as the given $n$th term in the Fibonacci sequence?

With a given $n$ and I am trying to find the value of $x$, as in: $$Fib(x)=n$$ Using the formula for Fibonacci sequence, where $\varphi$ is the Golden Ration ($\approx1.61803399\ldots$) $$Fib(z) = ...
2
votes
3answers
107 views

Proving no rational satisfy $p^2 = 2$

In Rudin's analysis example 1.1, he tried to show the following Let $A$ be the set of all positive rationals $p$ such that $p^2<2$ and let $B$ consist of all positive rationals $p$ such that $p^2 ...
4
votes
2answers
234 views

Did I do this proof right?

I am not sure if I did the proof right, so I wanted to see how most of you did this. I am trying to solve this problem: Let $x, y \in \mathbb N$ be relatively prime. If $xy$ is a perfect square, ...
6
votes
1answer
71 views

Prove that $\{\frac{\phi (n)}{n}\}_{n \in \Bbb N}$ is dense in $[0,1]$

suppose that $\phi(n)$ is Euler function. prove that, $\{\frac{\phi (n)}{n}\}_{n \in \Bbb N}$ is dense in $[0,1]$ (if $A_n=\{1 \leq m \leq n | m \in \Bbb N ; gcd(n,m)=1\}$ then $\phi(n)=|A_n|$) I ...
1
vote
1answer
82 views

Set of integer p-adics-Proposition

Proposition: "$\mathbb{Z}_p$ contains only the ideals $0$ and $p^n \mathbb{Z}_p$ for $n \in \mathbb{N}_0$. It holds $\bigcap_{n \in \mathbb{N}_0} p^n \mathbb{Z}_p=0$ and $\mathbb{Z}_p \ ...
1
vote
4answers
58 views

Is there numbers that don't fit in our sets of numbers?

It is said that the first numbers we used were natural numbers like $0$, $1$ ,$2$... in $\mathbb{N}$. Then we discovered negative numbers $-1$,$-2$... , and classified them all in $\mathbb{Z}$. Then ...
0
votes
1answer
71 views

On riemann zeta function [on hold]

What is the importance of the zeta functions at even and odd integers to prime numbers? Do these encode properties of primes like the zeros on $-1/2+ai$ line encodes or are they studied for their own ...
1
vote
2answers
43 views

Chinese remainder theorem other way around

I need to solve the following equation: $x^2\equiv 1 (\textrm{mod }1000)$ According to the chinese remainder theorem I can rewrite this as: $x^2 \equiv 1 (\textrm{mod }8)$ and $x^2 \equiv 1 ...
2
votes
1answer
44 views

Find the sum of digits of m?

Let $m$ be the number of numbers from set $\{1,2,3,\dots,2014\}$ which can be expressed as difference of the squares of two non negative integers. The sum of digits of $m$ is... My attempt: I ...
2
votes
1answer
26 views

When is it possible to find a relatively prime pair among $n$ numbers?

Suppose I have a set of $n$ numbers and their gcd is $d$. If I divide every number by $d$, is it possible to find a pair that is relatively prime? Intuitively yes, but how do I prove it? I tried ...
2
votes
0answers
19 views

How to prove there exists a positive integer $k,d$ such $\sum_{i=1}^{2k}a_{id}=k-2014$

Let $\{a_{n}\}_{n=1}^{\infty}$ be a non-negative integer such that: for any postive integer $m,n$, we have: $$\sum_{i=1}^{2m}a_{in}\le m.$$ Show that there exist positive integers $k,d$ such that: ...
10
votes
0answers
87 views

How find all postive integer number such $(n+k)\nmid \binom{2n}{n}$

Question: Find the all integer $k$,such there are exist infinitely many $n$ such $$(n+k)\nmid \binom{2n}{n}$$ This is china 2014 (CMO problem 4),it's have been end exam three hours ago. I ...
1
vote
0answers
36 views

How to prove taht a product of two complete residue system is not a complete residue system?

Claim. Let $n$ be a natural number and $A=\{0,1,2,3,\cdots,n-1\}$ be a complete set of residues modulo $n$. Let $\sigma$ be a permutation of $A$. Show that the set $C=\{\sigma(i)i:i\in A\}$ is not a ...
0
votes
0answers
26 views

Is there an online calculator in which you can type a number and have it tell you if it could be a Lychrel number or not?

Say you type 7326 into it, it runs a few calculations and tells you it reaches 99099 in three iterations. But if you type in a number like 887, it runs a reasonable number of iterations (say, twenty) ...
1
vote
1answer
51 views

primes of the form $4k+3$ and sums of squares [duplicate]

It is well-known that if $p$ is a prime of the form $4k+3$ and $p|x^2+y^2$ then $p|x$ and $p|y$. I forget what is the name of this result, and where can I find a proof (please provide a link).
1
vote
1answer
57 views

The ring is a principal ideal domain, especially an integral domain.

The following holds for the ring $ \mathbb{Z}_p, p \in \mathbb{P}$: The ring $ \mathbb{Z}_p $ is a principal ideal domain, especially an integral domain. I try to understand the following proof: ...
4
votes
1answer
60 views

Twin Primes between $n$ and $2n$

Is it theoretically possible for there to always be a twin prime pair between $n$ and $2n$ for all sufficiently large $n$ (assuming of course that there are infinitely many twin primes) or would this ...
1
vote
1answer
29 views

Sum of Coefficients and Number of Terms in Trinomials and Quadrinomials

I already know how to find the sum of coefficients in a binomial, but how do you do it for a trinomial/quadrinomial (after like terms are added)? Example Problem: $(wa+xb+yc+zd)^n$ (all variables are ...
2
votes
1answer
45 views

Some questions about prime divisors and number of primes

For an integer $n \ge 2$, let $\omega (n)$ denote the number of distinct prime divisors of $n$ and $\pi (n) $ be number of primes not exceeding $n$. Let $a_1, \ldots, a_k$ be integers greater than ...
21
votes
2answers
2k views

Fermat's Last Theorem (Case n = 3) Question

A very simple question. We all know that there are no solutions to $x^3 + y^3 = z^3$ for integer $x$, $y$ and $z$, $xyz\neq 0$, but are rational $x$, $y$ and $z$ possible? Thanks.
4
votes
1answer
74 views

What motiveted Gauss to formulate his theorem on quadratic reprocity?

Im trying to connect his work on quadratic reciprocity with some simple question, like solution to certain diophantine equation or representing primes. Any ideas? I find it hard to imagine that he out ...
7
votes
1answer
127 views

Proving that $T$:$(x_1,…,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},…,\frac {x_n+x_1}{2})$ leads to nonintegral components

Start with $n$ paiwise different integers $x_1,x_2,...,x_n,(n>2)$ and repeat the following step: $T$:$(x_1,...,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},...,\frac {x_n+x_1}{2})$ ...
-4
votes
0answers
69 views

Is 1+2+3+4+5+…=infinity? [duplicate]

Internet tells its -1/12 , but I think if you keep adding numbers it will be Infinity. Am I wrong? Or is it that me and the Internet is both correct, but then -1/12=infinity which is wrong.
4
votes
5answers
1k views

Theorem: the first positive number to have 500 divisors has to be even.

How can I get started on this proof? I was thinking originally: Let $ n $ be odd. (Proving by contradiction) then I dont know.
2
votes
3answers
72 views

Find all intergers such that $2n^2+1$ divides $n^3+9n-17$

Find all intergers such that $2n^2+1$ divides $n^3+9n-17$. Answer : $n=(2 \ and \ 5)$ I did it. As $2n^2+1$ divides $n^3+9n-17$, then $2n^2+1 \leq n^3+9n-17 \implies n \geq 2$ So $n =2$ is ...
-6
votes
1answer
58 views

A big challenge on Number theory [on hold]

Let $N=\frac{60^{2014}}{7}$. What is the sum of the first $2014$ digit before the decimal point of $N$?
2
votes
0answers
55 views

Prove that $\sum\limits_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2})$. [on hold]

Let $a,b\in\mathbb{Z}$, and $f\in C^2([a,b])$ such that $|f''(t)|\asymp \lambda$ for $a\le t\le b$. Prove that $$\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2}).$$ ...
5
votes
2answers
53 views

The rate of growth of small divisors of an integer

\begin{align} 1 & \times 360 \\ 2 & \times 180 \\ 3 & \times 120 \\ 4 & \times 90 \\ 5 & \times 72 \\ 6 & \times 60 \\ 8 & \times 45 \\ 9 & \times 40 \\ 10 & \times ...
3
votes
1answer
34 views

Bounds for general character sums over finite fields

Let $\mathbb{F}_q$ be a finite field with $q$ elements, let $\chi$ be the canonical additive character of $\mathbb{F}_q$, let $\psi$ be a non-trivial multiplicative character of $\mathbb{F}_q$, and ...
23
votes
0answers
168 views
+50

Is There An Injective Cubic Polynomial $\mathbb Z^2 \rightarrow \mathbb Z$?

Earlier, I was curious about whether a polynomial mapping $\mathbb Z^2\rightarrow\mathbb Z$ could be injective, and if so, what the minimum degree of such a polynomial could be. I've managed to ...
1
vote
2answers
50 views

Does $p_{1}^x + p_{2}^y = n$ have uniqe solution for $x$ and $y$ ($p_{1}, p_{2}$ are primes).

If I'm given a value $n$. And I know its of the form $p_{1}^x + p_{2}^y$, can I be sure that there is a unique solution for $x$ and $y$ and Can I determine values of $x$ and $y$, If I know the ...
2
votes
1answer
47 views

Solve $y^2=x^4+x+2$ in $x,y\in \mathbb{Z}$

I have to solve $y^2=x^4+x+2$ in $x,y\in \mathbb{Z}$ and right now I am stuck. This is how far I came: A little manipulation yields $y^2-2=x(x+1)(x^2-x+1)$. $x=1$ and $y=\pm 2$ are solutions. Assume ...
1
vote
0answers
41 views

Solve the Diophantine equation $y^3=4x^2+4x+5$ in $x,y\in\mathbb{Z}$

I have to solve the Diophantine equation $y^3=4x^2+4x+5$ where $x,y\in\mathbb{Z}$ and I have been thinking now for a long time and I have really no clue how to do this. The only hint given in the ...
1
vote
1answer
50 views

Solve $y^2=x^3-4$ in $x,y\in \mathbb{Z}$

I am having trouble solving the diophantine equation given in the title. This is how far I came: We can factor in $\mathbb{Z}[i]$ $y^2+4=x^3\Rightarrow (y+2i)(y-2i)=x^3$. I want to show now that ...
3
votes
1answer
38 views

Diophantine equation involving factorial …

Question . Find all positive integer solutions to the equation below , $$(n-1)!+1=n^m$$ (i)observe that $n>1$ and $n$ is a prime number (if not we can choose a prime number $p<n$ such that $p|n$ ...
-1
votes
2answers
23 views

Neat Diophantine Equation Question

After some fairly tedious work including studying multiple different cases separately, I have found all the solutions to $$a^n+1=b^2 $$ where $a$, $b$, $n$ can take on the value of any integer, be it ...
6
votes
0answers
64 views

When can $n^k+k$ be a perfect square?

For what positive integers $k$ does there exist a positive integer $n$ such that $n^k+k$ is a perfect square? Certainly for all $k$ such that $k+1$ is a perfect square, since we can substitute $n=1$. ...
23
votes
0answers
152 views

How many non-rational complex numbers $x$ have the property that $x^n$ and $(x+1)^n$ are rational?

In this answer, I proved the following statement: For each positive integer $n$, there are finitely many non-rational complex numbers $x$ such that $x^n$ and $(x+1)^n$ are rational. These complex ...
1
vote
0answers
32 views

What are some applications of parametrization of curves and surfaces?

I know that we can find all elements of a quadratic field with norm 1 by rational parametrization of conics, it can be used to show that some Diophantine equations are not so easy to solve, and that ...
2
votes
2answers
26 views

Classification of numbers on the base of binary representation

The problem is the following. I would like to find a simple algorithm or principle of classification of numbers regarding their presentation in binary form. Let's consider an example. The numbers by ...
9
votes
1answer
146 views

Source of Hardy-Littlewood's 2nd Conjecture

In what paper do Hardy and Littlewood first mention, specifically, their 2nd conjecture? It is not mentioned specifically in Partitio Numerorum III. This conjecture is usually expressed as ...
1
vote
2answers
63 views

About the infinitude of some kind of primes? [closed]

I will propose this proof: A Mersenne number always has the form $$2^{p}-1=4n+3$$ since for all $p≥2$ we have $$2^{p}-1≡-1(mod4)≡3(mod4)$$ The Dirichlet prime number theorem ...