Question

The primes 37,67,73,79,... are of the form p = 36ab + 6a - 6b + 1, with a >= 1, b >=1

Show that no pair of twin primes can contain a prime of this form.

Answer 1

Let p,q be twin primes. The difference would be
36(ab-a2b2)+6(a-a1)-6(b-b2)

Answer 2

The least difference is 6

Answer 3

No wait lol

Answer 4

Ok here again.. I'll fix my mistake

Answer 5

Looks bit tricky ...but easy :)

Ok let us assume that we have a twin prime of this form call that p, then p =36ab + 6a - 6b + 1 (for some a and b)

then since we assumed this guy p is a twin prime then p-2 or p+2 is prime .
Now , we are required to prove that both these will not be prime ie if p is a prime (of that form) then p+2 & p-2 are not-prime

Lets start

1) proving p+2 is composite(not-prime) whenever this guy p is prime .
seen easily by adding 2 see : p =36ab + 6a - 6b + 1 +2 =p =36ab + 6a - 6b + 3
which has a factor of 3 . [end of 1]

2) proving p-2 requires a bit of manipulation techniques . see if p =36ab + 6a - 6b +1
then p-2=p =36ab + 6a - 6b - 1 which can be factorized into (6b+1)*(6a-1)
since p=6a(6b+1)-(6b+1) =(6a-1)*(6b+1) [end of 2]

QED

Answer 6

p and q are twins assume p is of the form
q= 36ab + 6a - 6b +3 which is never prime xD

Answer 7

^ i agree :)

Answer 8

well no need to other way lol @epoweritheta already did it..

Answer 9

why what if in the twin primes pairs (ordered) <p,q> . you only proved p cannot be of the form 36ab + 6a - 6b + 1 .

but what if q is of such form ??? you just proved q+2 is not prime but you need to prove p is not a prime .

Answer 10

=) yeah i was gonna write the rest but u already did :P

Answer 11

:)