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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32 views

About an unpredictable sequence of primes [duplicate]

Let $p_n$ denote the sequence of prime numbers, with $p_0=2$. The obvious fact that the sequence $p_n$ is unpredictable is very known. I am asking if there is a mathematical proof for this. Or, this ...
1
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1answer
41 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
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1answer
23 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 ...
0
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0answers
11 views

How prove exsit postive integer $k,d$ such $\sum_{i=1}^{2k}a_{id}=k-2014$

let $\{a_{n}\}_{n=1}^{\infty}$ is non-negative integer,and such: for any postive integer $m,n$ have $$\sum_{i=1}^{2m}a_{in}\le m$$ show that: there exsit postive integer $k,d$ such ...
8
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0answers
45 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
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0answers
32 views

How prove The product of two complete residue system is not complete residue system?

Question: we know sets $A=\{1,2,3,\cdots,n-1,n\}$ is complete residue system modulo $n$, and other sets $B=\{a_{1},a_{2},\cdots,a_{n}\}$ is also complete residue system modulo $n$(such as ...
0
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0answers
23 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) ...
2
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0answers
31 views

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

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).
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0answers
39 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
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1answer
53 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 ...
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0answers
62 views

How do you find the square root of a binomial?

Mu Alpha Theta Nunn FL #25 Find the coefficient of the 4th term in the binomial expansion of $(9x-2y)^{\frac12}$. How I started: I first thought that there should be a $3\cdot\sqrt{x}$ term at the ...
1
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1answer
27 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 ...
0
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0answers
31 views

Some questions about prime divisors and no. of primes

Let for an integer $n \ge 2$ , $\omega (n)$ denote the no. of distinct prime divisors of $n$ and $\pi (n) $ be number of primes not exceeding $n$. Let $a_1,...,a_k$ be integers greater than $1$ and ...
15
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2answers
797 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
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1answer
70 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
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1answer
118 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: $T$:$(x_1,...,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},...,\frac {x_n+x_1}{2})$ ...
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0answers
67 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
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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.
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3answers
65 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 ...
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1answer
55 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
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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 $$\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2}).$$ ...
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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
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1answer
31 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 ...
15
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0answers
122 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
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2answers
47 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
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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 ...
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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
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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
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1answer
36 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 $p|n$ ...
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2answers
22 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
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0answers
62 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
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0answers
148 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
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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
142 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? [on hold]

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 ...
1
vote
1answer
18 views

Are Primitive Dirichlet Characters linearly independent.

For a positive integer $N$, let $$S_N=\{ \chi~\mid~ \chi \text{ is primitive Dirichlet characters modulo }F,\text{ where } F\mid N \}.$$ I want to check the Linear independence on $S_N$. More ...
6
votes
2answers
108 views

How to explain to a layman why Fermat's Last Theorem involves non-trivial math?

Fermat's Last Theorem states, given$$x^n + y^n = z^n$$ no three integers $x,y,z$ will satisfy the equation given integer value of $n$ greater than two. On the surface this seems like something that ...
0
votes
1answer
23 views

How to get the maximum and minimum number of length $m$ and the sum of the digits $s$

How to get the maximum and minimum of length $m$ and the sum of the digits $s$ By example: Length: 2 Sum of its digits: 15 Max: 96, Min: 69 Length: 2 Sum of its digits: 2 Max: 20, Min: 11
2
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1answer
33 views

Find $x$ such that $x \equiv7\pmod {37}$ and $x^2 \equiv 12\pmod {37^2}$

Find $x$ such that $x \equiv7 \pmod {37}$ and $x^2 \equiv 12\pmod {37^2})$ My attempt: Given $x \equiv7\pmod {37}$ so $37|(x-7)$ so $37^2|(x-7)^2$ so $x^2-14x+49 \equiv 0\pmod {37^2}$ as ...
6
votes
1answer
67 views

Number theory and abstract algebra question

So I was solving this question Find an isomorphism from the additive group $\mathbb Z_6$ to the multiplicative group of units $U_7$ in $\mathbb Z_7$. I found that $3$ is generator for U7 by brute ...
2
votes
3answers
52 views

Is there a number congruent to 1 modulo infinitely many primes?

Let $A=\left\{ p_{r},p_{r+1},\dots\right\}$ a (infinte) set of consecutive prime numbers (if you prefer, if $\mathfrak{P}$ is the set of all prime numbers, $A=\mathfrak{P}-\left\{ ...
0
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1answer
24 views

Hilbert class field whose class number is 1.

How to describe Hilbert class field of an imaginary quadratic field whose class number is 1 ? What happens to unramification at finite places ?
0
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1answer
26 views

Looking for methods for approximating an iterative equation regarding primes

In a previous question, I was looking for an equation for counting the number of the number of integers between $1$ and $x$ that have a prime factor besides $2$ or $3$. There were 2 iterative ...
2
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0answers
24 views

Decomposition of abelian varieties up to isogeny

Let $A_1,A_2,B_1,B_2$ be simple abelian varieties over a number field $k$. Suppose that $A_1\times A_2$ is $k$-isogenous to $B_1\times B_2$. Can we deduce that (up to reordering the factors) $A_1$ is ...
3
votes
1answer
89 views

Extending the zeta function to semiprimes, etc.

The Riemann Zeta function is defined for $s > 1$ as \begin{align} &\prod _{n=1}^{\infty}\dfrac{1}{1 -\ p_{n}^{\ \ -s}}\\ \end{align} It is possible to extend the zeta function to semiprimes ...
0
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1answer
42 views

Proving $m$ is prime when $a^{m-1}\equiv 1\pmod m$ and factors of $m-1$ satisy $a^n\equiv r\pmod m,r\neq1$

If $a^{m-1}\equiv 1\pmod m$, and all factors of $m-1$, say $n (n< m-1)$ satisfy $$a^n\equiv r\pmod m,r\neq1$$ then $m$ is a prime. I want to prove this proposition, but it is a little difficult ...
0
votes
1answer
19 views

Factorization with a Primitive Factor of Polynomials

Question: Let $f,g\in\Bbb Q[x]$. Why is it that $\rm\color{#c00}{(1)}$ if $f$ is monic then $f=\frac{1}{a}f^*$ for some primitive polynomial $f^*\in\Bbb Z[x]$ and $a\in\Bbb Z$ ? ...
4
votes
1answer
87 views

The Island in the Miracle Sea. (Christmas edition)

To all of you who love math like me, I have this puzzling riddle that I hope you find interesting : On Christmas Eve just after midnight, Santa was riding his sleigh over the Miracle Sea when ...
53
votes
3answers
732 views

A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational

Let $x$ be a real number and let $n$ be a positive integer. It is known that both $x^n$ and $(x+1)^n$ are rational. Prove that $x$ is rational. What I have tried: Denote $x^n=r$ and $(x+1)^n=s$ ...