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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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 ...
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2answers
15 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 ...
7
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1answer
69 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 ...
2
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1answer
37 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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1answer
173 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 ...
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0answers
33 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 ...
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103 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 ...
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1answer
47 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 ...
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1answer
32 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
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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 ...
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2answers
35 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 ...
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1answer
343 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 $s>1$ ...
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3answers
150 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
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1answer
803 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 ...
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3answers
678 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 $(x+1)^n=s$ ...
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4answers
168 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 ...
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0answers
24 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 ...
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1answer
24 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 ...
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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 ...
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4answers
137 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 ...
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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 ...
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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 ...
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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
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2answers
103 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 ...
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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 ...
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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) ...
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1answer
79 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 ...
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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 ...
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2answers
55 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 ...
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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
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1answer
31 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
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1answer
64 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 ...
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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\{ ...
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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 ...
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1answer
23 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 ?
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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 ...
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0answers
206 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 ...
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0answers
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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 ...
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4answers
119 views

Binary operation commutative, associative, and distributive over multiplication

Is there any binary operation that is commutative, associative, and distributive over multiplication? I asked this question in my head a while ago, and I posted it in various forums. However, having ...
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8answers
7k views

Can someone please explain the Riemann Hypothesis to me… in English?

I've read so much about it but none of it makes a lot of sense. Also, what's so unsolvable about it?
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2answers
281 views

Divisibility in a recurrent sequence

Let $a_1=0$, $a_2=\alpha$, and $a_n=\lambda a_{n-1}+\mu a_{n-2}$ for $n\geq 3$. Are there positive integers $\alpha$, $\lambda$, $\mu$ such that $$a_{p^2} \equiv 0 \mod p $$ for every prime ...
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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 $a\in\Bbb Z$ ? ...
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2answers
1k views

Why is $\varphi$ called “the most irrational number”?

I have heard $\varphi$ called the most irrational number. Numbers are either irrational or not though, one cannot be more "irrational" in the sense of a number that can not be represented as a ratio ...
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1answer
80 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 ...
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9answers
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How far can one get in analysis without leaving $\mathbb{Q}$?

Suppose you're trying to teach analysis to a stubborn algebraist who refuses to acknowledge the existence of any characteristic $0$ field other than $\mathbb{Q}$. How ugly are things going to get for ...
2
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1answer
36 views

How many distinct values of floor(N/i) exists for i=1 to N.

Say we have a function $F(i)=\text{floor}(N/i)$. Then how many distinct values of $F(i)$ will exist for all $0 \leq i \leq N$ e.g. We have $N=25$ then. $F(1)=25$ $F(2)=12$ $F(3)=8$ $F(4)=6$ ...
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0answers
83 views

Related to $\pi$ and $\tau$ - are these numbers transcendental, irrational, or rational?

Here are three numbers whose decimal expansions are listed in OEIS. Are they transcendental, irrational, or rational numbers? Note: $\tau = 2*\pi$ and the last two values are in radians. A233700. ...
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1answer
76 views

Minimum number of ways to color each integer

I have seen this problem floating around for a while but with no answer. Since the USAMTS deadline has passed, I would really like to see an answer for this. The farthest I got with this was that $n ...
7
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0answers
58 views

If $p\equiv 1,9 \pmod{20}$ is a prime number, then there exist $a,b\in\mathbb{Z}$ such that $p=a^{2}+5b^{2}$.

I have to prove that if $p\equiv 1,9 \pmod{20}$ is a prime number then there exist $a,b\in\mathbb{Z}$ such that $p=a^{2}+5b^{2}$. I consider the quadratic field $\mathbb{Q}(\sqrt{-5})$, with ring of ...
1
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1answer
145 views

Flexes of cubic curve

Which are the flexes of the cubic curve of Fermat $$x^3+y^3+z^3=0$$ at $\mathbb{P}^2(\mathbb{C})$ ? Could you give me a hint how we could find the flexes? Do we have to use maybe the following ...