# Tagged Questions

Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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### Can a conic over $\mathbb{Q}$ with no $\mathbb{Q}$-points have a point of degree 3?

Let $C$ be the conic in $\mathbb{P}^2$ given by $ax^2 + by^2 + cz^2 = 0$ with $a,b,c\in\mathbb{Q}$. (every genus 0 curve over $\mathbb{Q}$ can be given this way right?) Suppose $C$ has no rational ...
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### Does Euler's $\phi$ function have the same value in arbitrarily large subsets of $\mathbb{N}$?

As my most recent question still does not have any answers and it appears to be a difficult problem, I propose the following problem (that seems easier), but which I still could not manage to solve: ...
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### Prove that powers of any fixed prime $p$ contain arbitrarily many consecutive equal digits.

Prove that powers of any fixed prime $p$ contain arbitrarily many consecutive equal digits. It is an intuitive re-statement of Baltic Way 2012 (I think there are shortlists in Baltic Way every ...
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### How many values of $k$ satisfy $\left (\frac{k}{p}\right )=\left (\frac{k+1}{p}\right)=1$ where p is a odd prime?

The values of $k$ must be between $1$ and $p-1$ this means : $$k\in\left\{1,2,\cdots,p-1\right\}$$ The question: Given an odd prime $p$ What is the number of elements ...
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### Prove that every odd prime number can be written as a difference of two squares.

Prove that every odd prime number can be written as a difference of two squares. Prove also that this presentation is unique. Is such presentation possible if p is just an odd natural number? Can 2 ...
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### Number of Solutions to a Diophantine Equation

I am asked the following: Show that the number of integer solutions to $y^p=x^2+2$ for any odd prime $p$ is at most $p-1$. I checked that for $y^p=x^2+2$, the same method for $y^3=x^2+2$ works ...
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### A game with two dice

Imagine a game with two dice, played by two people and a referee. The referee rolls the first die and the number will determine the number of times that the second die will be rolled. The two players ...
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### Conjecture involving semi-prime numbers of the form $2^{x}-1$

Let $x$ be a positive integer such that $(2^{x}-1)=pq$ , where $p$ and $q$ are prime numbers. I want to show that either $p^{2} \bmod x \equiv 1$ or $q^{2} \bmod x \equiv 1$ (or both of course). Is ...
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### Non-integer bases and irrationality

I read somewhere: When it comes to properties like prime, irrational, rational, divisible by 2, etc., nothing changes when you change base. But I'm not sure about the rational/irrational one. ...
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### Period of decimal for $1/n$, odd part of $n+1$, and primes.

Let $n$ be a nature number is coprime to $10$,such that the period of the decimal expansion of $1/n$ is a divisor of $n-1$, and let $c$ be the "cycle length of $n$" (defined below). If $n-1=2^xc$ ...
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### Elements of finite order in the group of arithmetic functions under Dirichlet convolution.

Let $(G, ∗)$ be the group of arithmetic functions $f : N \to C$ that satisfy $f (1)\neq 0$, with group operation given by the Dirichlet product $∗$. The identity function $I$ is the identity element ...
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### Bounding the prime counting function

How can I get inequalities that bound the prime counting function if I have the following inequalities for some functions $f(x)$ and $g(x)$: $$g(x)<\psi(x)<f(x),$$ where $\psi(x)$ is second ...
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### A high-powered explanation for $\exp U(n)=2\iff n\mid24$?

In What's so special about the divisors of $24$? it is noted that the exponent of the group of units modulo $n$, that is the highest order of an element of $U(n):=(\Bbb Z/n\Bbb Z)^\times$, is ...
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### RSA: Creating a key of desired length

Thanks and with respect to the users of this site, I've succeeded in creating an Encryption/Decryption procedure for the RSA algorithm. I also implemented a Miller-Rabin probabilistic primality test. ...
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### Are limits on exponents in moduli possible?

Suppose I show that: $$x^{f(z)/g(z)} = y \pmod{4}$$ is impossible for some given positive integers $x$ and $y$, where, \begin{align*} f(z) &= \phi(4) k_1(z) + 1 \\ &= 2 k_1(z) + 1\\ g(z) ...
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### Proving $P( n) =n^{\phi(n)} \prod\limits_{d \mid n} \biggl(\frac{d!}{d^d} \biggr)^{\mu(n/d)}$

Actually, i had posted this long ago in MO, and i didn't get a reply to this question as it was unfit. Now, this is an exercise, in some textbook ( i think Apostol) and i would be happy if i can ...
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### Show that a Sophie Germain prime $p$ is of the form $6k - 1$ for $p > 3$

A Sophie Germain prime is a prime $p$ such that $2p + 1$ is also prime. According to a comment in OEIS A023212 (https://oeis.org/A023212), such a prime $p$ is of the form $6k - 1$ for $p > 3$. ...
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### Rational points on $y^2 = 12x^3 - 3$.

Prove by elementary arguments that the only rational point on the title curve is $(x, y) =(1,\pm 3)$. My attempt was the standard approach of factoring $(y+i\sqrt 3)(y - i\sqrt 3) = 12x^3$, but it ...
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### Proof of $\phi(n) = \sum_{n|d} \mu(d) \cdot\frac nd$

I'd like to prove $\phi(m)=\sum_{m|d}\mu(d)\cdot\frac md$. If I'm right then we have for euler-phi $\phi(n) = \sum_{m \leq n,\gcd(m,n)=1} 1$ Which means: as long as $m$ is less or equal than $n$ ...
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### Cubic diophantine equation

How can I solve the equation $x^3+x-1=y^2$ in positive integers? I know this equation defines an elliptic curve but this seems to be a non-elementary way to solve the question. Is there a more ...
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### Algorithm to find solutions $(p,x,y)$ for the equation $p=x^2 + ny^2$.

As the classical book of David Cox argues, Assume the conditions are satisfied and $p$ can be represented as $x^2 + ny^2$. What would be a way to find solutions $(p,x,y)$ efficiently? Ideally, one ...
Find all POSITIVE integer solutions to the following cubic equation: $x^3+2x+1=y^2$. Notice how the left side of the equation resembles $x^2+2x+1=(x+1)^2$. The only solutions I've been able to find ...
### sets with no asymptotical density over $\mathbb N$
Let's consider the Natural density on $\mathbb N$ defined by: Take $A\subset \mathbb N$; define the sequence $x_n= \dfrac{|A\cap[1,n]|}{n}$, and then if $\lim\limits_{n\to\infty} x_n$ exists, ...