Intereting Posts

$\lim_{n \to \infty} \mid a_n + 3(\frac{n-2}{n})^n \mid^{\frac1n} = \frac35$. Then find $\lim_{n \to \infty} a_n$.
The “Empty Tuple” or “0-Tuple”: Its Definition and Properties
There exists a vector $c\in C$ with $c\cdot b=1$
Solving $2x – \sin 2x = \pi/2$ for $0 < x < \pi/2$
Proof that the largest eigenvalue of a stochastic matrix is 1
A strong inequality from Michael Rozenberg
What is the most surprising result that you have personally discovered?
Proof of $\arctan{2} = \pi/2 -\arctan{1/2}$
Is the integral of a measurable function measurable?
If $ f(f(f(x)))=x$, does$ f(x)=x$ necessarily follow?
For $5$ distinct integers $a_i$, $1\le i\le5$, $f(a_i)=2$. Find an integer b (if it exists) such that f(b) = $9$.
Evaluate the following integral $\int_{0}^{10}\sqrt{-175e^{-t/4}+400}dt$
$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$
$\Delta u$ is bounded. Can we say $u\in C^1$?
Group theory proof of existence of a solution to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$

Let $a\in \mathbb{Z}$ be relatively prime to $p$ prime. Then show that the seqeunce $\{a^{p^{n}}\}$ converges in the $p$-adic numbers.

This to me seems very counter intuitive. Since $(a,p)=1$ the norm will always be $1$. I really have no idea what to do with $|a^{p^{n}}-a^{p^{m}}|$ factoring gets me nothing and we can’t use the nonarchimedian property because they have the same norm. Any hints or ideas would be great. Thanks.

- Quadratic Integers in $\mathbb Q$
- How to arrive at Ramanujan nested radical identity
- Are elementary and generalized hypergeometric functions sufficient to express all algebraic numbers?
- What are the integers $n$ such that $\mathbb{Z}$ is integrally closed?
- Show that $x^2 + x + 12 = 3y^5$ has no integer solutions.
- Ideals of $\mathbb{Z}$ geometrically

- How do you prove that $p(n \xi)$ for $\xi$ irrational and $p$ a polynomial is uniformly distributed modulo 1?
- Two exercises on characters on Marcus (Part 2)
- Why does the natural ring homomorphism induce a surjective group homomorphism of units?
- Maximize :: $A = B \times C$
- Finding all positive integer solutions to $(x!)(y!) = x!+y!+z!$
- Numbers of the form $\frac{xyz}{x+y+z}$, second question
- Are primes randomly distributed?
- Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} < 1$?
- Newman's “Natural proof”(Analytic) of Prime Number Theorem (1980)
- Congruence properties of $x_1^6+x_2^6+x_3^6+x_4^6+x_5^6 = z^6$?

Recall that the Euler totient function has values $\phi(p^n)=p^{n-1}(p-1)=p^n-p^{n-1}$ for all $n$. This means that for all $a$ coprime to $p$ we have the congruence

$$

a^{p^n}\equiv a^{p^{n-1}}\pmod{p^n}.

$$

By raising that congruence to power $p^{m-n}$ a straightforward induction on $m$ proves that

$$

a^{p^m}\equiv a^{p^{n-1}}\pmod{p^n}

$$

for all $m\ge n$. This holds for all $n$ implying that the sequence is Cauchy, and thus convergent w.r.t. the $p$-adic metric.

- How to show $\int_{0}^{\infty}e^{-x}\ln^{2}x\:\mathrm{d}x=\gamma ^{2}+\frac{\pi ^{2}}{6}$?
- Power (Laurent) Series of $\coth(x)$
- Associativity of product measures
- Why can't $\sqrt{2}^{\sqrt{2}{^\sqrt{2}{^\cdots}}}>2$?
- Every power series expansion for an entire function converges everywhere
- If f is continuous does this means that the image under f of any open set is open?
- How many ways are there to arrange the letters in the word “mississippi” such that all “p” precede all “i”?
- Sums of complex numbers – proof in Rudin's book
- Which translation to read of Euclid Elements
- a formula involving order of Dirichlet characters, $\mu(n)$ and $\varphi(n)$
- River flowing down a slope
- Embedding of a field extension to another
- Prove Euler's Theorem when the integers are not relatively prime
- For a polynomial $p(z)$ with real coefficients if $z$ is a solution then so is $\bar{z}$
- The accuracy from left to right and that from right to left of the floating point arithmetic sums