Intereting Posts

A(nother ignorant) question on phantom maps
Fourier transform for dummies
Group of even order contains an element of order 2
If $4^n + 2^n + 1$ is prime, then $n$ must be a power of $3$
Is this a valid proof of $\lim _{n\rightarrow \infty }(1+\frac{z}{n})^n=e^z$?
Norm of powers of a maximal ideal (in residually finite rings)
One-to-One correspondence in Counting
Why is every positive integer the sum of 3 triangular numbers?
Prove that the empty relation is Transitive, Symmetric but not Reflexive
Problem proving: $V = \ker T \oplus \operatorname{im}T$
Primitive recursive definition of the “divisibility” relation
Contour Integration to evaluate real Integral when there is no singularity
Integer solutions to $2x^2+5x+y^2=19$
Ramanujan Class Invariant $G_{125}$ and $ G_{5}$
What's the thing with $\sqrt{-1} = i$

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.

- Showing $\mathbb{Z}+\mathbb{Z}\left$ is a Euclidean domain
- minimal polynomial of power of algebraic number
- The resemblance between Mordell's theorem and Dirichlet's unit theorem
- Conclude that $\mathbb{Z}$ is not a UFD.
- Intuition and Stumbling blocks in proving the finiteness of WC group
- Need the norm of positive number be positive?

- Repeatedly taking mean values of non-empty subsets of a set: $2,\,3,\,5,\,15,\,875,\,…$
- Ideals in a Quadratic Number Fields
- Do these series converge to the von Mangoldt function?
- What is known about these arithmetical functions?
- Find all integers satisfying $m^2=n_1^2+n_1n_2+n_2^2$
- Higher dimensional analogues of the argument principle?
- What is the importance of the Collatz conjecture?
- modular form -Petersson inner product
- Prove: For odd integers $a$ and $b$, the equation $x^2 + 2 a x + 2 b = 0$ has no integer or rational roots.
- What are Diophantine equations REALLY?

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.

- show that $f^{(3)}(c) \ge 3$ for $c\in(-1,1)$
- Change basis so that a positive definite matrix $A$ is now seen as $I$.
- Is a compact simplicial complex necessarily finite?
- Why is polynomial regression considered a kind of linear regression?
- Which rings containing the complex field are, as vector spaces over that field, isomorphic to $\mathbb{C}^2$?
- Find the limit $\lim_{x \to 1} \left(\frac{p}{1-x^p} – \frac{q}{1-x^q}\right) $ $p ,q >0$
- Sum of an infinite series of fractions
- How to count the number of solutions for this expression modulo a prime number $p$?
- Sum of truncated normals
- What are logarithms?
- Fatou's lemma and measurable sets
- Does this condition on the sum of a function and its integral imply that the function goes to 0?
- Why does $\sum_{n=0}^\infty (\sin{\frac{1}{n}})(\sinh{\frac{1}{n}})(\cos n)$ converge
- If $\phi(\alpha)$ is prime in $\mathbb{Z}$, show that $\alpha$ is prime in $\mathbb{Z}$
- Existence of irreducible polynomials over finite field