If p##\geq##5 is a prime number, show that p^{2}+2 is composite?

In summary, the proof shows that if p is a prime number greater than or equal to 5, then p^2+2 is a composite number. This is because p can only take the forms 6k+1 or 6k+5, which results in p^2+2 being divisible by 3. This can be shown using modulo arithmetic, where p^2+2 is congruent to 3 modulo 6. Therefore, the proof is correct and valid.
  • #1
Math100
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Homework Statement
If p##\geq##5 is a prime number, show that p^{2}+2 is composite.
[Hint: p takes one of the forms 6k+1 or 6k+5.]
Relevant Equations
None.
Proof: Suppose p##\geq##5 is a prime number.
Applying the Division Algorithm produces:
p=6k, p=6k+1, p=6k+2, p=6k+3, p=6k+4 or p=6k+5 for some k##\in\mathbb{Z}##.
Since p##\geq##5 is a prime number,
it follows that p cannot be divisible by 2 or 3 and p must be odd.
Thus, p takes one of the forms 6k+1 or 6k+5.
Now we consider two cases.
Case #1: Suppose p=6k+1.
Then we have p^{2}+2=(6k+1)^{2}+2
=36k^{2}+12k+3
=3(12k^{2}+4k+1)
=3m,
where m=12k^{2}+4k+1 is an integer.
Thus, p^{2}+2 is composite.
Case #2: Suppose p=6k+5.
Then we have p^{2}+2=(6k+5)^{2}+2
=36k^{2}+60k+27
=3(12k^{2}+20k+9)
=3m,
where m=12k^{2}+20k+9 is an integer.
Thus, p^{2}+2 is composite.
Therefore, if p##\geq##5 is a prime number, then p^{2}+2 is composite.

Above is my proof for this problem. Can anyone please verify/review to see if this is correct?
 
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  • #2
It looks fine. Slicker is
$$p = \pm 1 \ ( mod \ 6) \ \Rightarrow \ p^2 +2 = 3 \ (mod \ 6)$$$$\Rightarrow \ p^2+2 = 3k$$
 
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  • #3
PeroK said:
It looks fine. Slicker is
$$p = \pm 1 \ ( mod \ 6) \ \Rightarrow \ p^2 +2 = 3 \ (mod \ 6)$$$$\Rightarrow \ p^2+2 = 3k$$
Thank you.
 
  • #4
You should definitely be using modulo arithmetic more. It's not like it's something that takes a lot to learn. It's the sort of thing that some people use before they realize it's a thing that gets taught in maths classes.
 
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Related to If p##\geq##5 is a prime number, show that p^{2}+2 is composite?

1. What is a prime number?

A prime number is a positive integer that is only divisible by 1 and itself. In other words, it has exactly two factors.

2. How can you prove that p##\geq##5 is a prime number?

One way to prove that a number is prime is by using the Sieve of Eratosthenes, which involves eliminating all multiples of numbers up to the square root of the given number. If the number cannot be eliminated, then it is prime.

3. What does it mean for a number to be composite?

A composite number is a positive integer that has more than two factors. In other words, it is divisible by at least one number other than 1 and itself.

4. How does p^{2}+2 being composite prove that p##\geq##5 is a prime number?

If p^{2}+2 is composite, then it must have more than two factors. Since p^{2}+2 is equal to p^{2}+1+1, this means that p^{2}+2 is divisible by both p and 1+1, which means p is not the only factor. Therefore, p^{2}+2 being composite proves that p is not a prime number.

5. Are there any exceptions to this rule?

Yes, there is one exception to this rule. If p=3, then p^{2}+2=11, which is a prime number. However, this is the only exception, as all other values of p##\geq##5 will result in a composite p^{2}+2.

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