All Questions
26
questions
2
votes
0
answers
68
views
Largest possible prime factor for given $k$?
Let $k$ be a positive integer.
What is the largest possible prime factor of a squarefree positive integer $\ n\ $ with $\ \omega(n)=k\ $ (That is, it has exactly $\ k\ $ prime factors) satisfying the ...
- 85.1k
5
votes
1
answer
161
views
Infinitely many primes with $2$ and $3$ generating the same set of residues
Prove that there are sets $S$ and $T$ of infinitely many primes such that:
For every $p\in S$ there exists a positive integer $n$ such that $p\mid 2^{n} - 3$.
For every $p \in T$ the remainders mod $...
- 2,823
1
vote
1
answer
396
views
Using Euler's Totient Function, how do I find all values n such that, $\varphi(𝑛)=14$
I just recently started working with Euler's Totient Function, and I came across the problem of solving for all possible integers $n$ such that $\varphi(n)=14$. I know there are similar questions with ...
- 99
7
votes
1
answer
117
views
On numbers with small $\varphi(n)/n$
Let $\Phi(n) = \varphi(n)/n = \prod_{p|n}(p-1)/p$ be the "normalized totient" of $n$.
Some facts:
$\Phi(p) = (p-1)/p < 1$ for prime numbers with $\lim_{p\rightarrow \infty}\Phi(p) = 1$
$\Phi(n) = ...
- 18.2k
8
votes
0
answers
182
views
Odd numbers with $\varphi(n)/n < 1/2$
The topic was also discussed in this MathOverflow question.
From $\varphi(n)/n = \prod_{p|n}(1-1/p)$ (Euler's product formula) one concludes that even numbers $n$ must have $\varphi(n)/n \leq 1/2$ ...
- 18.2k
0
votes
1
answer
163
views
On prime-perfect numbers and the equation $\frac{\varphi(n)}{n}=\frac{\varphi(\operatorname{rad}(n))}{\operatorname{rad}(\sigma(n))}$
While I was exploring equations involving multiple compositions of number theoretic functions that satisfy the sequence of even perfect numbers, I wondered next question (below in the Appendix I add a ...
user243301
0
votes
1
answer
195
views
Finishing the task to find the solutions of $\frac{1}{x}-\frac{1}{y}=\frac{1}{\varphi(xy)},$ where $\varphi(n)$ denotes the Euler's totient function
In this post I evoke a variant of the equations showed in section D28 A reciprocal diophantine equation from [1], using particular values of the Euler's totient function $\varphi(n)$. I ask it from a ...
user243301
11
votes
7
answers
3k
views
Only finitely many $n$ such that $\phi(n) = m$
Let $\phi(n)$ be Euler's totient function.
How do I show that there are only finitely many such $n$ with $\phi(n) = m$, for each positive integer $m$?
I've written $n$ as a product of primes; $n = ...
- 2,482
0
votes
3
answers
167
views
About the solutions of $x^{\varphi(yz)}+y^{\varphi(xz)}=z^{\varphi(xy)}$, being $\varphi(n)$ the Euler's totient
In this post we denote the Euler's totient function as $\varphi(n)$ for integers $n\geq 1$. I wondered about the solutions of the equation
$$x^{\varphi(yz)}+y^{\varphi(xz)}=z^{\varphi(xy)}\tag{1}$$
...
user243301
3
votes
2
answers
145
views
Euler's Totient Function: $\phi(n)\geq n\cdot 2^{-r}$.
My friend's teacher made a list with this problem: If $n$ has $r$ distinct prime factors, show that:
$$\phi(n)\geq n\cdot 2^{-r}$$
I tried to help her, but I am not very good in number theory
- 65
4
votes
2
answers
308
views
On questions involving the radical of an integer and different number theoretic functions: the Euler's totient function
We denote for an integer $n>1$ its square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$
with the definition $\operatorname{rad}(1)=1$. You can see this ...
user243301
2
votes
0
answers
55
views
On variations of Erdős squarefree conjecture: presentation and a question as a simple case
I'm inspired in the so-called Erdős squarefree conjecture, this section from Wikipedia, to state in this post a question, involving a different arithmetic function, that due its difficulty I feel as ...
user243301
2
votes
1
answer
68
views
On miscellaneous questions about perfect numbers II
Let $\varphi(m)$ the Euler's totient function and $\sigma(m)$ the sum of divisors function. We also denote the product of primes dividing an integer $m>1$ as $\operatorname{rad}(m)$, that is the ...
user243301
4
votes
1
answer
331
views
Markov triples that survive Euler's totient function
I'm inspired in a recent post of this MSE. We denote the Euler's totient function in this post as $$\varphi(n)=n\prod_{p\mid n}\left(1-\frac{1}{p}\right).$$
Suppose we have three positive integers $a,...
user243301
2
votes
2
answers
152
views
The arithmetic function $\frac{\operatorname{rad}(2n)}{n+\varphi(n)+1}$ and a characterization of twin primes
We denote the Euler's totient function as $\varphi(n)$, and the radical of the integer $n> 1$ as
$$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\ p\text{ prime}}}p,$$
taking $\operatorname{rad}(1)...
user243301