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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9
votes
1answer
130 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 ...
23
votes
0answers
140 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 ...
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.
1
vote
2answers
43 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
334 views

Theorems similar to Euler's theorem ($a$, $n$ are not coprime)

It is well known that if $\gcd(a,n)=1$, then $a^{ϕ(n)}=1$ mod $n$. Are there any results similar to Euler's theorem that can be used when $a$ and $n$ are not coprime. Feel free to add any ...
5
votes
3answers
151 views

If $(m,n)\in\mathbb Z_+^2$ satisfies $3m^2+m = 4n^2+n$ then $(m-n)$ is a perfect square.

I came across this question on another forum. The question is: $$ \text{If $m,n\in \mathbb{Z}_+$ such that $3m^2+m=4n^2+n$, then $(m-n)$ is a perfect square.}$$ I have managed to partially prove ...
1
vote
3answers
106 views

how to show $\mathbf{Q} $ is not free

we know torsion free plus finitely generated $\rightarrow$ free and that $\mathbf{Q}$ is torsion free is easy. But how to show Q is not finitely generated? and not free?
2
votes
3answers
57 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 ...
1
vote
0answers
76 views

$a^{N(\mathcal{P})} \equiv a \pmod{\mathcal{P}}$

Let $\mathcal{P}$ a prime ideal in $\mathcal{O}_K$. Show that $a^{N(\mathcal{P})} \equiv a \pmod{\mathcal{P}}$, $a \in \mathcal{O}_K$. I have this: $N(\mathcal{P})=p^f$, $f$ is the inertia degree of ...
2
votes
0answers
65 views

If $x \sim U(Z_n^*)$ then $x^2 \pmod n\sim U(QR_n)$?

Define: $Z_n^*=\{x \in Z_n | \operatorname{gcd}(x,n)=1\}$ $QR_n=\{x \in Z_n | \exists r \in Z_n \; s.t. \; r^2 =x\}$ How can I show that $x \sim U(Z_n^*) \implies x^2 \pmod n \sim U(QR_n)$? Thank ...
5
votes
2answers
52 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 ...
-4
votes
0answers
66 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.
0
votes
2answers
55 views

Why is it so difficult to use geometric methods to construct weight one modular forms?

I am trying to understand why it is that geometric methods of constructing modular forms of weight one for $SL(2,\mathbb Z)$ fail. I have a relatively rudimentary understanding of it, but it would be ...
3
votes
1answer
28 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 ...
-5
votes
1answer
51 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 $N$?
8
votes
1answer
196 views

How to solve $y^2=3x^4+3x^2+1$ for integers.

If $x,y \in \mathbb Z$ , then find all the solutions of $$y^2=3x^4+3x^2+1$$ I was asked this question by my friend who said that he encountered this while solving another problem. I have ...
1
vote
0answers
52 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 $$\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2}).$$ ...
6
votes
0answers
59 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 $n=1$. ...
1
vote
2answers
115 views

Looking for a more efficient primality testing Algorithm than Miller-Rabin

I am looking for a practical probabilistic primality testing algorithm that is more superior than Miller-Rabin. By "more superior", I mean that the probability of giving the wrong answer is better ...
9
votes
0answers
86 views

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 ...
2
votes
1answer
44 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
40 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
49 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
35 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
21 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 ...
11
votes
1answer
345 views

A simple way to obtain $\prod_{p\in\mathbb{P}}\frac{1}{1-p^{-s}}=\sum_{n=1}^{\infty}\frac{1}{n^s}$

Let $ p_1 <p_2 <\ldots <p_k <\ldots $ the increasing list in set $\mathbb{P}$ of all prime numbers . By Infinite geometric series $\sum_{k=0}^\infty r^k = \frac{1}{1-r}$ for all $s>1$ ...
2
votes
3answers
153 views

What is the maximum difference between two successive real numbers in the given floating point representation?

The following is a scheme for floating point number representation using 16 bits. Sign :- Bit 15 Exponent:-Bit 14-9 Mantissa :- Bit 8-0 Let $s, e,$ and $m$ be the numbers represented in binary in ...
31
votes
1answer
806 views

Is $\sqrt1+\sqrt2+\dots+\sqrt n$ ever an integer?

Related: Can a sum of square roots be an integer? Except for the obvious cases $n=0,1$, are there any values of $n$ such that $\sum_{k=1}^n\sqrt k$ is an integer? How does one even approach such ...
53
votes
3answers
719 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 $(x+1)^n=s$ ...
3
votes
4answers
174 views

Is there any known relationship between Goldbach's comet G(n) and the prime counting function (${\pi(n)}$)?

The "extended" Goldbach conjecture defines R(n) as the number of representations of an even number n as the sum of two primes, but the approach is not related directly with ${\pi(n)}$, is there any ...
0
votes
0answers
28 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 ...
0
votes
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
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 ...
0
votes
1answer
41 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 ...
4
votes
2answers
122 views

How prove for any $k$ then have $a^2_{1}+a^2_{2}+a^2_{3}+\cdots+a^2_{k}=m^3$

for any positive integer $k$,there exsit $m\in N$ and $a_{i}\in N,i=1,2,\cdots,k$,such (1): $a_{i}\neq a_{j},i\forall i\neq j$, (2): $$a^2_{1}+a^2_{2}+a^2_{3}+\cdots+a^2_{k}=m^3$$ My idea: if ...
1
vote
2answers
57 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 ...
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 ...
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 ...
4
votes
1answer
62 views

Fairly good semiprime estimate

I have found a nice estimate for the semiprime counting function \begin{align} &f_{2}(x):=x \log \left( \log (x)/\log \left( a+a/ \exp\left( (\log (\log (x)-2)-1)^2/2\right) (\log (x)-2) \right) ...
3
votes
1answer
85 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 ...
13
votes
1answer
133 views

Natural density of solvable quintics

A recent question asked about the topological density of solvable monic quintics with rational coefficients in the space of all monic quintics with rational coefficients. Robert Israel gave a nice ...
1
vote
2answers
56 views

A Generalization of Carmichael Numbers

Obviously, from Fermat's Little Theorem, the condition of $p$ being prime is equivalent to there being some number $a$ of multiplicative order $p-1$ mod $p$. Moreover, this is equivalent to saying ...
0
votes
1answer
20 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
votes
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
66 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
50 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\{ ...
3
votes
2answers
40 views

Eigenvalues of a unimodular matrix

Let $U$ be a unimodular matrix, i.e. $U \in \mathbb{Z}^{n \times n}$, and $\text{det}(U) = \pm 1$. Do the real (or complex for that matter) eigenvalues of $U$ admit a special structure? Edit: It is ...
0
votes
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 ?
4
votes
2answers
568 views

Card Shuffling [SPOJ]

The original question is posted on SPOJ, and included below: Here is an algorithm for shuffling N cards: 1) The cards are divided into K equal piles, where K is a factor of N. 2) The ...
11
votes
0answers
218 views
+50

On the problem of polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$

The question titled "Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$" which was posed on MathOverflow attracted quite a lot of attention (and may be the question with most wrong ...