Question

For a positive integer n, if both n and n + 2 are prime, then they are known as twin primes. For example, 59 and 61 are twin primes. Whether or not there are an infinite number of twin primes is a famous unsolved problem in number theory.

Find all positive integers n such that n, n + 2, and n + 4 are all prime.

Answer 1

Hint: Try looking at them mod 3.

Answer 2

well it follows that \(n,n+2,n+4\) must be a sequence of odd numbers... one such (immediate) pair is given by \(1,3,5\), another is \(3,5,7\), ...

Answer 3

If n is a prime congruent to 1 mod 3, will n+2 and n+4 still be prime? What about if n is congruent to 2?

Answer 4

oops \(1\) is not prime...

Answer 5

@joemath314159 is correct... considering \(\mod3\) is natural because the range of our values is \(4\)

Answer 6

@joemath314159 is always correct:)

Answer 7

Let n be a prime such that \(n\equiv 1\pmod{3}\) Then \(n+2\equiv 1+2\equiv 0\pmod{3}\) and we see that n+2 isn't prime (its divisible by 3). So n cant be equivalent to 1 modulo 3. Now we try \(n\equiv 2\pmod{3}\). It follows that \(n+4\equiv 2+4\equiv 0\pmod{3}\), so n+4 isnt prime. So n can't be equivalent to 2 modulo 3. That leaves \(n\equiv 0\pmod{3}\), but the only prime that is equivalent to 0 modulo 3 is 3 itself. Hence if n, n+2, and n+4 are all primes, then n must equal 3. This gives 3, 5, and 7 as the only primes that satisfy the condition.

Answer 8

consider that \(n\equiv0,1,2\mod 3\) hence \(n=3k,3k+1,\text{ or }3k+2\). if \(n=3k\) then \(n\) is not prime unless \(k=1\). if \(n=3k+1\) then \(n+2=3k+3=3(k+1)\) is not prime unless \(k+1=1\). if \(n=3k+2\) then \(n+4=3k+6=3(k+2)\mod 3\)

Answer 9

damnit @joemath314159 i wrote that out earlier but I went out to help my dad

Answer 10

lolol, my bad >.< I'm a little busy too, im packing.

Answer 11

ill medal the guy who doesn't get metaled

Answer 12

We metal'd each other :P

Answer 13

thats love

Answer 14

and respect :) oldrin's answers are spot on and interesting.

Answer 15

unless \(k+2=1\)... and then it remains to be shown that \(k=0\) gives prime \(3(k+1)\) but nonprime \(3k\) and similarly \(k=-1\) gives prime \(3(k+2)\) but nonprime \(3k\)... it must be that \(k=1\) hence \(n=3,n+2=5,n+4=7\)

Answer 16

hehe I try

Answer 17

I really should log off thought lol, I gotta finish packing >.<

Answer 18

yeah , I like watching you two work.

Answer 19

when do you leave joe?

Answer 20

Monday o.O

Answer 21

o wow, and what part of texas?

Answer 22

its gonna be soooooooooooooooooooooooooooooooo hot

Answer 23

I'm moving to College Station. Inbetween Austin in Houston

Answer 24

nice, I was born in tyler......so hot

Answer 25

Yeah, its 103 here in San Antonio right now >.<

Answer 26

ouch :(

Answer 27

ahh I didnt realize you were in texas now

Answer 28

well at least log on and let us know you got there safely.

Answer 29

Sure thing :) I'm out, see you all later.

Answer 30

jeez and I thought Florida was too warm

Answer 31

Augghhh!!! Who to give the medal to?

Answer 32

@joemath314159

Answer 33

i lived in florida al well. its got nothing on texas imo

Answer 34

imo, Florida is worse, because its super humid. It might not get as hot, but at least its dry hot here lolol.

Answer 35

yeah true

Answer 36

and joemath is going to say oldrin.bataku

Answer 37

this is my life :)

Answer 38

@orple8 pick one, and ill give the other

Answer 39

Thanks :)

Answer 40

i dont think either of them care anyway:)

Answer 41

yeah I mostly just enjoy doing the problems

Answer 42

true. :)