Congruence Class Proofs: Tips and Examples

• MHB
• Joe20
In summary, the conversation discusses a question about examining cases in $\mathbb{Z}_3$ and finding a counterexample. It is suggested to write the numbers in a simpler format, which leads to the conclusion that $a^2 + b^2 = 0$ and a and b must be less than 3. The purpose of $3(3n^3+3m^3+2an+2bn)= 0$ is to show that it can be ignored "modulo 3". This equation also helps to determine that a and b must be positive integers.
Joe20
Hi, I have tried the question as attached. I am not sure if I am correct. You help is greatly appreciated. Thanks in advance!

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Hi Alexis87,

I like the idea to examine each of the cases. A few were left out - e.g., $[a]==[2]$ - and I don't know if that was intentional on your part or not. It may be a little neater to note that the result of squaring in $\mathbb{Z}_{3}$ can only be $[0]$ or $[1]$, in which case there are only three possibilities to check. Your counterexample looks good.

I would consider it simpler to write the first number as [a]= 3n+ a and the second as = 3m+ b where a and b are one of 0, 1, or 2. Then $[a]^2+ ^2= (3n+a)^2+ (3m+ b)^2= 9n^2+ 6an+ a^2+ 9m^2+ 6bm+ b^2= 3(3n^3+3m^3+ 2an+ 2bm)+ a^2+ b^2= 0$.

So we must have a and b less than 3 and $a^2+ b^2= 0$. From that, a= 0, b= 0 so that [a]= = 0.

HallsofIvy said:
I would consider it simpler to write the first number as [a]= 3n+ a and the second as = 3m+ b where a and b are one of 0, 1, or 2. Then $[a]^2+ ^2= (3n+a)^2+ (3m+ b)^2= 9n^2+ 6an+ a^2+ 9m^2+ 6bm+ b^2= 3(3n^3+3m^3+ 2an+ 2bm)+ a^2+ b^2= 0$.

So we must have a and b less than 3 and $a^2+ b^2= 0$. From that, a= 0, b= 0 so that [a]= = 0.

Hi HallsofIvy,

may I ask what's the purpose of showing this expression:
=3(3n3+3m3+2an+2bm)+a2+b2=0 ?

Is there any thing we can interpret from this equation?

So for "So we must have a and b less than 3" does it mean that because it is Z subscript 3?

The purpose is to realize that $3(3n^3+ 3m^3+ 2an+ 2bn)$ is a multiple of 3 and so can be ignored "modulo 3". That is why I can say "We must have a and b positive integers less than 3 such that $a^2+ b^2= 0$.

1. What is a congruence class?

A congruence class is a set of all numbers that have the same remainder when divided by a given modulus. For example, in the congruence class of 4 mod 7, all numbers that have a remainder of 4 when divided by 7 belong to this class (e.g. 11, 18, 25, etc.).

2. What is a congruence class proof?

A congruence class proof is a method of proving that two numbers are congruent (have the same remainder when divided by a given modulus) by showing that they belong to the same congruence class. This is often done using algebraic manipulations and the properties of congruence.

3. What are some tips for writing a congruence class proof?

Some tips for writing a congruence class proof include: clearly stating the given numbers and modulus, using the properties of congruence (such as addition, subtraction, and multiplication) to manipulate the numbers, and clearly showing each step of the proof.

4. Can you provide an example of a congruence class proof?

Sure! Let's prove that 15^2 is congruent to 1 mod 7. First, we can rewrite 15^2 as (14 + 1)^2. Using the distributive property, this becomes 14^2 + 2(14)(1) + 1. Next, we can simplify 14^2 to 196, which is congruent to 0 mod 7. Therefore, the entire expression becomes 0 + 2(14)(1) + 1, which is congruent to 1 mod 7. This shows that 15^2 is indeed congruent to 1 mod 7.

5. What are some real-life applications of congruence class proofs?

Congruence class proofs are used in various fields of mathematics, such as number theory and abstract algebra. They can also be applied in computer science, particularly in cryptography and coding theory. Additionally, congruence classes have practical applications in fields such as engineering and physics, where they are used in solving equations and modeling systems.

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