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CROWDMATH 2024: Where Does the Goldbach Conjecture Hold?

Where Does the Goldbach Conjecture Hold? Polymath project forum
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Where Does the Goldbach Conjecture Hold? Polymath project forum
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exercise 1.3
aeemc2   2
N Nov 3, 2024 by aeemc2
One of the irreducible criteria is well-known for polynomials with integers coefficients and also holds naturally for polynomials of $\mathbb{N}_0[x]$ is Eisenstein's irreducible criterion that states
Criterion 1 (Eisenstein's irreducible criterion).
Let $p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0 \in \mathbb{N}_0[x] $. If there exists a prime number $p$ satisfying the following three conditions:

- $p$ divides each $a_i$ for $0 \leq i < n$.
-$p$ does not divide $a_n$, and
-$p^2$ does not divide $a_0$.

Then the polynomial $p(x)$ is irreducible.

We have another criterion, which is even easier to check.

Criterion 2: Let $p(x) \in  \mathbb{N}_0[x]$. Suppose that $p(0) \neq 0$ and $p(1)$ is a prime number, then $p(x)$ is irreducible.

Proof
Indeed, suppose that $p(x)=q(x)r(x)$ for some $q(x), r(x) \in    \mathbb{N}_0[x]$. Note that $p(1)=q(1)r(1)$, and since $p(1)$ is prime, we know that either $q(1)=1$ or $r(1)=1$. WLOG, assume that $r(1)=1$, then $r(x)=x^n$ for some $n \in \mathbb{N}_0$. Note that if $n > 0$, then $r(0)=0$, and then $p(0)=0$, which contradicts the hypothesis. Therefore $r(x)=1$, and this implies that $p(x)$ is irreducible.

2 replies
aeemc2
Nov 1, 2024
aeemc2
Nov 3, 2024
A New CrowdMath Project (with Resource 0 and Some Exercises)
felixgotti   2
N Oct 15, 2024 by felixgotti
Hi everyone!

A New Research Project for CrowdMath 2024 has been released! This new research project will be about semidomains satisfying the statement of the Goldbach Conjecture.

We have the great pleasure to have Dr. Harold Polo with us, providing his direct help with this project, which is in turn motivated by the current research carried out by Dr. Polo in the intersection of semidomains and the statement of the Goldbach Conjecture.

Resource 0 (with some initial exercises) has just been posted. As always, I hope you enjoy learning the new material and working on the exercises. We plan to post the first open problem by next week.

Don't hesitate to bring your questions, solutions, and ideas to the message board!

Best,
Felix
2 replies
felixgotti
Sep 1, 2024
felixgotti
Oct 15, 2024
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