Modular Arithmetic: Solving Equations with 22x^2 = 11 mod 13

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In summary, the conversation discusses a method for solving equations involving modular arithmetic, specifically the equation 22x^2 = 11 mod 13. The method involves defining a variable z and using the equations az = c mod m and z = x^2 mod m to find the solutions without having to try all possibilities. It is mentioned that this method is useful for larger numbers and can save time compared to trial and error. The term "modular asthmatic" is also mentioned in a humorous context.
  • #1
rashida564
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Homework Statement
22x^2 + 13(x + 2y)^3 = 11 has no integer solutions x and y
Relevant Equations
Module asthmatic.
I am thinking of taking modular of 13 to both sides of equation. So it will become
22x^2 is 11 mod 13.
And try all the values from x equal to zero to x equal to 11.
Is their easier way to solve it
 
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  • #2
I think it is a good idea, what I would do is to define ##z=x^2## to have the equation ##az=c \pmod{m}## which is very easy to solve without trying an error (which is not a problem for this small numbers, but its always nice to have a method to solve things), and then you only need to solve ##z = x^2 \pmod{m}## which again can be done without trying an error.
 
  • #3
How can I find that there's no solution using your method, there's no integer solution
 
  • #4
Well, as I said, the equations ##ax=b \pmod{m}## and ##x^2=a \pmod{m}## are very common and important and you can find the solutions of both without trying and error (if you have not studied this then it's OK). Of course, that means that, if they don't have a solution you can prove it without having to try all the possibilities. (As I said, since you are working ##\pmod{13}## try and error is not a big deal, but if you were working at ##mod{97}## it would be really long, and a number with 10 digits would be impossible except maybe for a computer.)
 
  • #5
"Modular asthmatic"? Doesn't modular arithmetic have enough problems without being asthmatic as well?
 

What is modular arithmetic?

Modular arithmetic is a branch of mathematics that deals with numbers and their remainders when divided by a specific number, known as the modulus. It is often used to analyze and solve equations involving cyclical patterns or periodic phenomena.

What does "22x^2 = 11 mod 13" mean?

This notation means that the remainder of 22x^2 divided by 13 is equal to 11. In other words, when 22x^2 is divided by 13, the remainder will be 11.

How do you solve equations with modular arithmetic?

The first step in solving equations with modular arithmetic is to identify the modulus. Then, use the properties of modular arithmetic, such as the distributive property and rules of exponents, to manipulate the equation and isolate the unknown variable on one side. Finally, use trial and error or modular inverse to find the solution.

What is the purpose of solving equations with modular arithmetic?

Solving equations with modular arithmetic can help in understanding patterns and cycles in data and in finding solutions to problems that involve repeating or cyclical elements. It is also used in cryptography and computer science to ensure secure communication and data storage.

What are some applications of modular arithmetic?

Modular arithmetic has various applications in different fields such as number theory, cryptography, computer science, and coding theory. It is used in the development of encryption algorithms, error-correcting codes, and in analyzing patterns in data sets. It is also used in clock arithmetic and calculating leap years in calendars.

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