Pairs of twin primes
Twin primes may occur in pairs  i.e. 11, 13, 17, 19. A cursory check seems to indicate that they have to be of the form 90k + 11, 13, 17, 19. Has this ever been proven? If so has it ever been proven that the set of k's is infinite or is it finite?

Re: Pairs of twin primes
A less cursory check might throw up 5,7,11,13.

Re: Pairs of twin primes
Sorry  I meant after the single digits. The case you described is the only one where a number ending in 5 could appear.

Re: Pairs of twin primes
1481, 1483, 1487, 1489 is the first counterexample. (1491 = 41 + 90 * 16)
However what is true is that they are all of the form: 30k + 11, 13, 17, 19. This can easily be proven by supposing we have primes n+11,n+13,n+17,n+19 (with n nonnegative). n must be even because otherwise n+11 is even and therefore not prime. So 2n. If [itex]n \equiv 1\pmod 3[/itex], then 3 divides n+17 which is a contradiction. If [itex]n \equiv 2\pmod 3[/itex], then 3 divides n+11 which is a contradiction. Thus 3n. If [itex]n \equiv 1\pmod 5[/itex], then 5 divides n+19 which is a contradiction. If [itex]n \equiv 2\pmod 5[/itex], then 5 divides n+13 which is a contradiction. If [itex]n \equiv 3\pmod 5[/itex], then 5 divides n+17 which is a contradiction. If [itex]n \equiv 4\pmod 5[/itex], then 5 divides n+11 which is a contradiction. Thus 5n. We now have 2*3*5=30n. 
Re: Pairs of twin primes
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Re: Pairs of twin primes
Of 165 occurences of twin prime pairs taken from primes in the range 101000000 there are 60 of the form 90k+11,13,17,19. That's slightly more than you would predict from the 30k+11,13,17,19 constraint mentioned in rasmhop's post, but not surprisingly so.

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