Dirichlet theorem in algebra
I am not a good combinatorics solver so I did spend more time working on some combinatorics problems when I was in the team in secondary school. One of the most famous theorems that maybe everyone has encountered when studying this field is the Dirichlet theorem. I will discuss a deeper look into it with some complex problems that I have studied, solved, found, and collected.
We may all know how the problem of 3 rabbits and 2 cages works: There exists a cage that has at least 2 rabbits. In a more “professional” way, Dirichlet Theorem can be said:
Given that A and B are two finite sets; m and n are, respectively, the number of elements in both sets. If there exists a k number (k>0) such that m > k.n and we set a consistency between pairs of elements from both sets. There exist k+1 elements of A that are consistent with elements of B.
Easy problems can be told like: there are 50 students in a class. Prove that there are at least 5 students who have the same month of birth. The problem in answering Dirichlet problems is the difficulty of realizing what quantities are the “rabbits” and what are the “cages”. Some problems hide these quantities and you will have to reveal them yourselves. That would be a difficult job.
In this part, I will give some problems related to algebra that use Dirichlet:
Easy problems can be told like: there are 50 students in a class. Prove that there are at least 5 students who have the same month of birth. The problem in answering Dirichlet problems is the difficulty of realizing what quantities are the “rabbits” and what are the “cages”. Some problems hide these quantities and you will have to reveal them yourselves. That would be a difficult job.
In this part, I will share with you a problem related to algebra that use Dirichlet:
At first glance, this seems to be a difficult problem as proving a sum as irrational numbers is pretty challenging. However, you can use counter-proof: assume that there is no real number x such that these numbers are all irrational numbers. Therefore, there must be at least 1 rational number.
The most challenging thing in this problem is that we have to create our cages and rabbits. See you in the next part where I would dig deeper into other fields.