- #1

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Prove the following statements:

**1)**If

**p≥5**is a prime number, then

**p[itex]^{2}[/itex]+2**is a composite number.

Attempt:

I know that any prime number

**p>3**will have the form of either

**6k+1**or

**6k-1**, and so I am able to put

**6k-1**into the equation

**(6k-1)[itex]^{2}[/itex]+2**

**36k[itex]^{2}[/itex]-12k+1+2**

**3(12k[itex]^{2}[/itex]-4k+1).**

Similarly I substitute

**6k+1**into the same equation with the result

**3(12k[itex]^{2}[/itex]+4n+1)**

thus showing that

**p[itex]^{2}[/itex]+2**is indeed a composite number.

Have I gone about this the right way?

**If**

2)

2)

**a**and

**8a-1**are prime, then

**8a+1**is composite.

Attempt:

I have been stuck on this one as I'm not really sure where to begin. The one thing I can think of doing is starting with a counter example.

Suppose

**a**and

**8a-1**are composite, this tells me (I think) that there is d|a such that

**1<d<a**and also

**c|8a-1**such that

**1<c<8a-1**, this is about as far as I have come and I'm not really sure how to proceed.

Any sort of help or hints would be great.

Thanks!