Number Theory Questions: Proving p and x2 Congruencies

In summary, the conversation is about two math questions that the person is having trouble with. The first question involves proving that the prime divisors of an integer have the form 6k+1. The second question involves finding a solution for a congruence problem. The person has received some hints but is still struggling and is seeking more help.
  • #1
ak_89
5
0
I have a few questions I am having troubles with. If someone can push me in the right direction that would be awesome. Here are the questions:

1. Prove that the prime divisors, p cannot equal 3, of the integer n2-n+1 have the form 6k+1. (Hint: turn this into a statement about (-3/p) )

2. Show that if p is congruent to 1 (mod 4), then x2 is congruent to -1 (mod p) has a solution given by the least residue (mod p) of ( (p-1)/2)!

I honestly have no idea how to start. I would greatly appreciate some help.
Thanks
 
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  • #2
For the first question, try multiplying the n2-n+1 term by some integer and rearranging things...

Unfortunately, I'm stumped on how to do the second question :uhh:
 
  • #3
Thanks! I got that proof. But I am still stuck on the second question as well. I played around with it.. but I have yet to get anywhere that is useful to prove the question.

I could really use some help.
 
  • #4
Hint for #2: Use Wilson's Theorem.

Petek
 

1. What is number theory?

Number theory is a branch of mathematics that deals with the properties and relationships of numbers, particularly integers.

2. What is a congruence in number theory?

A congruence in number theory is a relationship between two numbers where the remainder when divided by a specific number is the same. For example, a ≡ b (mod m) means that a and b have the same remainder when divided by m.

3. How do you prove p congruency in number theory?

To prove p congruency, you must show that p is the same remainder when divided by a specific number, usually denoted by m. This can be done using modular arithmetic, which involves finding the remainder of a number when divided by m.

4. How do you prove x2 congruency in number theory?

To prove x2 congruency, you must show that x2 is the same remainder when divided by a specific number, usually denoted by m. This can also be done using modular arithmetic, by finding the remainder of x2 when divided by m.

5. What are some real-world applications of number theory?

Number theory has many real-world applications, including cryptography, coding theory, and computer science. It is also used in fields such as physics and engineering for solving problems involving integers and their properties.

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