4 Primes in 9 consecutive Integers This is not so much a question, as it is an interesting (to me at least) observation.
Part of a more complex problem involved finding primes in 9 consecutive Integers greater than 2.
Not being the smartest math guy out "there", I used my current math knowledge + logic, and quickly removed all even numbers as well as numbers ending in 5.
Left were numbers ending in 1, 3, 7 & 9.
I scribbled down some code in Python, and noticed that every time I found 4 primes in 9 consecutive integers, the lowest one always ended in 1.
I concentrated on numbers divisible by 3 and found that the lowest prime MOD 3 had to equal 2:

Remembering that if the sum of digits is divisible by 3, then the number itself is too, I logically discovered that the sum MOD 3 would increase by 2 when going from a number ending in 9 to one ending in 1, hence any sequence of "prime candidates" starting with a number ending in anything by 1 would contain all 3 possible results of the number MOD 3 = [0,1,2], whereas only a sequence starting with a number ending in 1, would only contain 2 of those, and therefore be able to have a sequence with all 4 numbers being prime candidates.
This may be extremely elementary for most of you math experts out there, but I just found it interesting, and discoveries like these are why I love math.
I would love to hear more facts about this particular "primes in 9 consecutive integers" study.
Thanks.
 A: It's not hard to show that any 'quadruplet primes' of the type described in the post are congruent (smallest to largest) to $11, 13, 17, 19$ $\pmod{30}$, with the exception of some quadruplets where the smallest prime is less than or equal to $5$.
If this sort of thing is interesting, one can think about double quadruplet primes where the smallest primes in each quadruplet differ by $30$.  The smallest such grouping (I think) is $1006301, 1006303, 1006307, 1006309, 1006331, 1006333, 1006337, 1006339$.
A: You are basically looking for $2$ consecutive pairs of twin primes, which is the only way for $9$ consecutive integers to contain $4$ primes larger than $3$.
It is not known whether there are infinitely many pairs of twin primes, so it is obviously not known whether there are infinitely many consecutive pairs of twin primes.
Of course, except for the first set ($5,7,11,13$), every such set will contain exactly one number ending with each digit ($1,3,7,9$).
There is nothing "too special" about it, as the only reason for this is the use of the decimal base system for representation.
