- #1
Amcote
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This is my first time posting anything on the forum so I apologize if I do anything wrong. I have enrolled myself into elementary number theory thinking we would be taught how to do proofs however it is apparently expected that we already know how to do this. And so since I am a beginner at this, it would be very nice for someone to point out whether I'm doing/presenting the problems incorrectly or inefficiently.
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?
2) If 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!
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?
2) If 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!