regularngon said:Homework Statement
Find all primes p (as a congruence) such that 11^((p-1)/2) = 1 modp
The Attempt at a Solution
I'm new to congruences and I don't really know to approach this. Any help greatly appreciated!
Prime numbers are unique and have special properties that make them useful in mathematical calculations. Solving congruence with prime numbers can help in identifying patterns and relationships between numbers, leading to a better understanding of number theory.
To solve congruence equations involving prime numbers, you can use various methods such as the Chinese Remainder Theorem, Fermat's Little Theorem, or Euler's Theorem. These methods involve manipulating the congruence equation to simplify it and find the solution.
This equation is known as the Euler Criterion and is used to determine whether a number is a quadratic residue (a number that has a square root modulo p) or not. It plays a crucial role in solving congruence equations involving prime numbers.
Yes, solving congruence with prime numbers has various applications in fields such as cryptography, coding theory, and computer science. Prime numbers are also used in determining the validity of credit card numbers and generating secure passwords.
While prime numbers have many useful properties, they can also be challenging to work with, especially when dealing with large numbers. Additionally, solving congruence with prime numbers may not always lead to unique solutions, and there may be multiple solutions to a congruence equation.