Intereting Posts

Showing $R$ is a local ring if and only if all elements of $R$ that are not units form an ideal
Suppose $1\le p < r < q < \infty$. Prove that $L^p\cap L^q \subset L^r$.
if $AB\neq 0$ for any non zero matrix $B$ then $A$ is invertible
The existence of a group automorphism with some properties implies commutativity.
Must perpendicular (resp. orthogonal) lines meet?
Given K balls and N buckets what is the expected number of occupied buckets
Smoothness of a real-valued function on $\mathbb{R}^n $
Intertwiners and $\text{SL}(2, \mathbb{F}_q)$, vector space decomposition of $\mathbb{C}\{X\}$?
Elements in $\hat{\mathbb{Z}}$, the profinite completion of the integers
Meaning of Rays in Polar Plot of Prime Numbers
prove that $(1 + x)^\frac{1}{b}$ is a formal power series
Characterization of hierarchically clustered graphs
Is there a function with infinite integral on every interval?
A (non-artificial) example of a ring without maximal ideals
What do $\pi$ and $e$ stand for in the normal distribution formula?

If $\phi(n)$ divides $n-1$, prove that $n$ is a product of distinct prime numbers (such as number is also called square-free, as it is divisible by no square greater than $1$).

- Pennies on a checkerboard.
- Dirichlet's theorem on primes in arithmetic progression
- Does $\lfloor(4+\sqrt{11})^{n}\rfloor \pmod {100}$ repeat every $20$ cycles of $n$?
- Is '10' a magical number or I am missing something?
- A theorem about prime divisors of generalized Fermat numbers?
- Why is $f(x) = x\phi(x)$ one-to-one?
- Completion and algebraic closure commutable
- About a mysterious sequence who seems to follow some patterns
- Find the common divisors of $a_{1986}$ and $a_{6891}$
- How do I get a sequence from a generating function?

If $\displaystyle n=\prod_ip_i^{e_i}$, then $\displaystyle\phi(n)=\prod_i(p_i-1)p_i^{e_i-1}$

Therefore,

$$

\frac{n-1}{\phi(n)}\prod_ip_i^{e_i-1}(p_i-1)=n-1

$$

If any $e_i\gt1$, then $p_i^{e_i-1}$ divides both $n$ and $n-1$. Contradiction.

Hint: If $p^2|n$ then $p|\phi(n)$. Can $\phi(n)|n-1$ in that case?

- Difficult limit problem involving sine and tangent
- Improved Betrand's postulate
- Area of a triangle in terms of areas of certain subtriangles
- Limits of recurrently defined sequences.
- We can define the derivative of a function whose domain is a subset of rational numbers?
- Is $\pi \cdot 7$ actually $22$?
- Coin flipping probability game ; 7 flips vs 8 flips
- If $2$ divides $p^2$, how does it imply $2$ divides $p$?
- Proving $f'(1)$ exist for $f$ satisfying $f(xy)=xf(y)+yf(x)$
- Product of spheres embeds in Euclidean space of 1 dimension higher
- Finite Sum $\sum\limits_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}$
- expected number of cards drawn exactly once (with replacement)
- Does Pi contain all possible number combinations?
- How to solve $4\sin \theta +3\cos \theta = 5$
- Construct a bijection between $\mathbb{Z}^+\times \mathbb{Z}^+$ and $\mathbb{Z}^+$