Problem with prime and composite numbers

In summary, the conversation discusses a proof for the statement that for any prime p greater than or equal to 5, p^2+2 is a composite number. The proof involves showing that p can be written in the form 6k+1 or 6k+5, and then demonstrating that (6k+1)^2+2 and (6k+5)^2+2 are both composite. The justification for this first step is also discussed, with the conclusion that every prime number greater than 5 can be written in the form 3k+1 or 3k+2.
  • #1
canningdevin
3
0
If p >= 5 is prime, prove that p^2 + 2 is composite.

So i noticed if we divide any p >= 5 by 6 we only get remainders of 1 or 5.

6 | 5 , r = 5
6 | 7 , r = 1
6 | 11, r = 5
6 | 13, r = 1
6 | 17, r = 5 and so on

so for my proof i am saying for p >= 5, p = 6k + 1 or 6k = 5

so for the first , p^2 + 2 = (6k+1)^2 + 2 = 36k^2 +12k + 3 = 3(12k^2 + 4k + 1)

and similarly, p^2 + 2 = (6k+5)^2 + 2 = 36k^2 + 60k + 27 = 3(12k^2 + 20k + 3)

so both of these are composite so that is great.

But is there a way i can justify that any prime p >= 5 can be written as 6k + 1 or 6k + 5?

I mean i can go through a ton of examples and the remainders are only 1 or 5 but examples don't mean anything.

Can someone help me justify that first step.
 
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  • #2
I just thought of something can someone tell me if this is correct. for p primes p >=5 we can only get remainders 1 or 5 when dividing by 6 because if we get 2 3 4 we see what happens

6k + 2 = 2(3k +1)
6k + 3 = 3(2k +1)
6k + 4 = 2(2k + 2) and none of these can be prime because they have factors of either 2 or 3.
Is this correct?
 
  • #3
Clearly, every prime number greater than 5 can be written in the form 3k+1 or 3k+2. Then all that you need to demonstrate is that (3k+1)2+2 and (3k+2)2+2 are composite. This is similar to your line of thinking.
 
  • #4
ah ok, thank you very much
 

What are prime and composite numbers?

Prime numbers are positive integers that are only divisible by 1 and themselves. Composite numbers are positive integers that have more than two factors.

Why is it important to understand prime and composite numbers?

Prime and composite numbers are fundamental concepts in mathematics and are used in many fields, including cryptography, computer science, and number theory. They also help us understand the properties and relationships between different numbers.

How can you identify prime and composite numbers?

A number is prime if it has exactly two factors, 1 and itself. A number is composite if it has more than two factors. To identify prime numbers, you can use the Sieve of Eratosthenes or perform a trial division.

What are some common misconceptions about prime and composite numbers?

One common misconception is that 1 is a prime number. However, 1 is only divisible by itself, so it does not meet the definition of a prime number. Another misconception is that all even numbers are composite. However, 2 is a prime number and the only even prime number.

What are some real-world applications of prime and composite numbers?

Prime and composite numbers are used in cryptography to create secure codes and protect sensitive information. They are also used in algorithms for data encryption and decryption. In computer science, they are used in data compression and error-correcting codes. In number theory, prime numbers are studied to understand their distribution and patterns.

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