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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:

Finding 4 Primes in 9 consecutive Integers

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.

Ole Drews Jensen
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    You're basically looking for two consecutive pairs of twin primes. This is the only way for $9$ consecutive integers to contain $4$ primes. So I suggest that you simply Google consecutive pairs of twin primes. – barak manos Oct 27 '15 at 14:13
  • I know this could probably be found from the Google search you suggested (I looked briefly), but is it correct that the smallest of the $4$ consecutive primes has a $1$ in the units place? I can see that $5, 7, 11, 13$ work, but anything larger? – CSCFCEM Oct 27 '15 at 14:32
  • Good catch! - Missed that one, which only works because 5 is a prime. However, I believe the above proves that any numbers above have to start with a prime ending in 1. – Ole Drews Jensen Oct 27 '15 at 14:38
  • @CSCFCEM, it's not just consecutive primes, as those can start with any of the "endings" [1,3,7,9], but consecutive primes in 9 consecutive integers. – Ole Drews Jensen Oct 27 '15 at 14:41
  • @OleDrewsJensen, Yea reading what I wrote I see I did not mention the 9 consecutive integer part (but I was thinking it!! I promise!). – CSCFCEM Oct 27 '15 at 14:54

2 Answers2

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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 $1006301, 1006303, 1006307, 1006309, 1006331, 1006333, 1006337, 1006339$.

paw88789
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  • Thanks, and you are absolutely correct:

    lastp=11

    for p in range(11,100000000000,10):

    ... if kingole.isprime(p):

    ... if kingole.isprime(p+2):

    ... if kingole.isprime(p+6):

    ... if kingole.isprime(p+8):

    ... if p-lastp==30:

    ... print(lastp,p)

    ... lastp=p

    ...

    1006301 1006331

     – Ole Drews Jensen Oct 27 '15 at 14:50
  • (sorry, unable to format a comment) – Ole Drews Jensen Oct 27 '15 at 14:52
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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.

barak manos
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  • Thanks for your input.

    Again, this may not be "anything too special" for a lot of you guys, but I found it pretty interesting. I didn't pay too much attention to a lot of math back in the stoneage when I went to school, so the majority of my math knowledge now is from self-study, experimentation, and trial-and-error.

     – Ole Drews Jensen Oct 27 '15 at 14:43
  • @OleDrewsJensen: You're welcome. – barak manos Oct 27 '15 at 14:45