Amannequin said:Homework Statement
If p is a prime and p>3, show that pr[itex]\equiv[/itex]1,5,7 or 11 (mod12)
Homework Equations
The Attempt at a Solution
Do I go about this by knowing that any prime p greater than 3 is of the form 6n+1 or 6n+5? Any direction on how to go about this will be helpful. Thanks.
"Mod" stands for "modulo" and refers to the remainder after dividing a number by another number. In this case, we are looking at the remainder when dividing prime numbers by 12.
This is because the statement only holds true for prime numbers greater than or equal to 3. Smaller prime numbers do not satisfy the given equation.
This statement can be proven using mathematical induction. First, we can show that it holds true for the base case p=3. Then, we can assume that it is true for p=k and use this assumption to prove that it is also true for p=k+1. By repeating this process, we can prove that it holds true for all prime numbers greater than or equal to 3.
This statement has important implications in number theory, specifically in the study of prime numbers. It shows that all prime numbers greater than or equal to 3 can be expressed as 1, 5, 7, or 11 when divided by 12. This can also be used to prove certain theorems and make predictions about the behavior of prime numbers.
Yes, this statement can be generalized to other moduli. For example, we can prove that all prime numbers greater than or equal to 5 satisfy pr ≡ 1, 3, 7, or 9 (mod 20). However, the specific values of 1, 5, 7, and 11 in the original statement are specific to the modulus of 12.