7:55 AM
@Mathphile Hi

8:45 AM
@TheSimpliFire Hi

2 hours later…
10:47 AM
@Peter why don't we prove all prime are odd and prove dirichlet theorem?
why is it 13 page long?
I mean why proof is 13 page long

1) 2 is an even prime 2) That all primes (except 2) are odd does not prove that every arithmetic progression an+b with coprime a,b produces infinite many primes. This is a deep result and to be honest, I never understood the proof. Why do you think that there is a trivial proof ?

11:22 AM
@Peter Ah sorry I forgot to exclude 2. Ok now my brain is activated. Yes saying all prime(except 2) are odd doesn't prove an+b since not every odd are prime. I was thinking there is trivial proof but I was wrong.
I want to understand the elementary proof.
they say this is easiest proof there but I don't even understand the notation. Also I don't know how this exponent is even derived.
Now if I ignore this I have another desicion that is to construct R from Q where R is real number and Q is rational.
Is it hard to construct this?

1 hour later…
12:44 PM
@Stupidquestioninc No, elementary does not mean easy. It means the actual techniques use involve basic analytic number theory but the construction is by no means easy

The reals can be "constructed" with the rationals using Cauchy-sequences , every real number is then an equivalence class of sequences converging to the same value. But this is a quite abstract construction. Note that there are even uncomputable real numbers, whose decimal expansion cannot be computed with an algorithm.
@TheSimpliFire Hi, what do you think about my Carmichael-numebrs above ?

I have no idea

Using caucy sequence sounds good but by equivalence class of sequence converging to the same values you mean their relationship is converging to same values?
May be I need to review some real analysis. I kinda forgot how caucy sequence converge to irrational.

1:00 PM
The easiest example is $\sqrt{2}$. The sequence $a_1=1$ , $a_{n+1}=\frac{a_n^2+2}{2a_n}$ for $n\ge 1$ converges towards $\sqrt{2}$

@Peter Thanks! Can you explain equivalence relation I kinda got confused by it.

Two sequences are equivalent if their difference tends to $0$ , but this is not actually important. Important is that every limit of a converging sequence of rational numbers is a real number.
And such a real number can be (theoretically) written as an infinite decimal expansion, but the digit-sequence can usually not be fully described. Even for $\sqrt{2}$ , there is no known "pattern" in the digits.

1:43 PM
@Peter Now I remember this from past real analysis course.
Reminds me of $\sqrt=1+1/(2+1/(2+1/...))$
1+x=2+1/(1+x)
@Peter How do you remember all these stuff. I hardly remember anything I learn,
I even forget definitions from group theory. And when I review it takes tons of time.