Number Theory Problem of the day

[Steven G]

Please show that p, p+2 and p+4 can't all be primes.
It's trivial, if you have the correct ammunition. I'd like to see how one would prove this without this special ammunition. Remember, have fun with this.

 

[Harry_the_cat]

What if p=3? Then you have 3, 5 and 7 which are all primes. Disproven!

 

[lev888]

What if p=3? Then you have 3, 5 and 7 which are all primes. Disproven!

Google finds this problem with an additional p>3 restriction.

 

[lev888]

Please show that p, p+2 and p+4 can't all be primes.
It's trivial, if you have the correct ammunition. I'd like to see how one would prove this without this special ammunition. Remember, have fun with this.

Spoiler

Assuming p>3.
If p is 3n it's not a prime.
If p is 3n+1, then p+2 = 3n+3 is not a prime
If p is 3n+2, then p+4 = 3n+6 is not a prime

 

[khansaheb]

Spoiler

Assuming p>3.
If p is 3n it's not a prime.
If p is 3n+1, then p+2 = 3n+3 is not a prime
If p is 3n+2, then p+4 = 3n+6 is not a prime

I think the proof should start with a statement like:

Every positive integer can be expressed in one of following forms: n or (n+1) or (n+2). In think Asimov's book (One, two, three,.....infinity) uses this logic (trick).

 

[topsquark]

I think the proof should start with a statement like:

Every positive integer can be expressed in one of following forms: n or (n+1) or (n+2). In think Asimov's book (One, two, three,.....infinity) uses this logic (trick).

I have a lot of respect for Asimov's non-fiction series...

but that one was Gamow's book!

-Dan

 

[khansaheb]

I have a lot of respect for Asimov's non-fiction series...

but that one was Gamow's book!

-Dan

That is absolutely correct - Gamow of "Alpher, Bethe, Gamow" fame. That book ("One, Two, three,...infinity") was my favorite book to give away to my teen-age, friends. Took me ~50 readings to rally understand everything in there (even the jokes)......