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I was flipping the math book where I saw a table of primes. The primes were marked in black bold.

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It's interesting to see that except 3,13,23, lots of primes (not necessarily consecutive) occur in pairs $p$ and $p+10$, and their distribution compare to other isolated primes didn't seem to reduce in $1000$. So I googled and there was a thing called https://en.wikipedia.org/wiki/Twin_prime .

My question was that was the pair of primes of module 10 just a coincidence?

Tito Piezas III
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    it may have something to do with the adoption of decimal system by humans, and the fact that these pairs each form a group of two bold numbers in the book, just one above the other, that makes them conspicuous, at least on the couple of pages that you have copied (and the corresponding initial segment of positive integers). – Mirko Apr 20 '18 at 03:33
  • @Mirko Thanks! I got really suspicious when those bold numbers were always close to/or a part of twin primes. Then I thought about prime's distribution and it didn't add up so I got confused. –  Apr 20 '18 at 03:36
  • I may not follow. Do you mean there are a lot of cases when a twin prime, like 821, also goes with a prime distance 10 from it, like 811, or even a distance multiple of 10 from it, like 761? All primes in this table (except for row one) are in columns headed 1,3,7,9, and given these primes are not too big, there are enough many of them to give the impression that these primes form a pattern, column 1,3 making twin primes conspicuous, similar for column 7,9 (and on the other hand pair 269,271 inconspicuous), and then numbers in the same column are same mod 10. Maybe just pigeon-hole principle. – Mirko Apr 20 '18 at 03:51
  • @Mirko I had a random pick and 100549 and 100559 were primes and 99367 and 99377 were primes etc. https://primes.utm.edu/lists/small/10000.txt ...they were not disappearing. –  Apr 20 '18 at 03:55
  • 99367 99371 ...102181 102191...103801 103811...e.t.c. where ever you look, you could find them.in the last 10 primes before 104730, there were 104683 104693 and 104701 104711. it just seemed to simply assume that its not true. (There were infinite twin primes, were there infinite $(p,p+10)$ pairs? –  Apr 20 '18 at 03:57
  • I had a random pick from the same table and picked 97729, 97771 and 97777 , neither twin, not mod 10. Answer to comment below: Actually 97777 , 97787, 97789 (but this is not random, since as I said, my random pick was 97729, 97771 and 97777, and then you purposefully picked 97777 , 97787, instead of say 97673, 97687, 97711, 97729 :) – Mirko Apr 20 '18 at 04:01
  • @Mirko actually 97777 97787 –  Apr 20 '18 at 04:02
  • @Mirko: I've added a table below. – Tito Piezas III Apr 20 '18 at 04:03
  • @TitoPiezasIII I copied your table, and added a few rows, as well as a column (the result of an initial error in my code). – Mirko Apr 20 '18 at 05:33
  • You might have a look at my answer https://math.stackexchange.com/a/1698703/1714 to a question which looks much similar to this one, but more detailed. – Gottfried Helms Apr 20 '18 at 09:40

2 Answers2

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(Too long for a comment.)

Actually, if humans had $12$ fingers and generally used the duodecimal system (base $12$), the results would have been more striking.

I used Mathematica to find the number $N$ of prime pairs $p$ and $p+m$ for the first $10000$ primes and the results are summarized below:

$$\begin{array}{|c|c|c|} \hline m&N&\text{name}\\ \hline 2&1270&\text{twin primes}\\ 4&1264&\text{cousin primes}\\ 6&\color{blue}{2538}&\text{sexy primes}\\ 8&1303&-\\ 10&1682&-\\ 12&\color{blue}{2515}&-\\ 14&1546&-\\ \hline \end{array}$$

The fact that all primes $p>3$ have form $6n\pm1$ may explain the preference for $m=6$ and $m=12.$

P.S. The interesting name "sexy prime" has to do with the Latin word for six (sex), though whoever coined it may have had other things in mind.

Tito Piezas III
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    As pointed out by Mirko, the table shown by the OP has $10$ columns since it was made by a $10-$fingered human, and highlighting the phenomenon seen by the OP. However, if it would have been better if it had $6$ or $12$ columns. – Tito Piezas III Apr 20 '18 at 04:15
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Here is my version of the table that Tito Piezas III presented in his answer. Initially I wrote the wrong code (using computer algebra Reduce) and wondered why I didn't get the same numbers as in Tito's table, then I realized that I was computing the number of pairs, for the first 10000 primes $p\ge3$ such that the next prime was a given distance $m$ from $p$ (e.g., distance $2,4, ...,36$). This number of primes is in column $K$ in the table below (whereas column $N$ is the number of pairs, for the first 10000 primes $p\ge3$ such that $p+m$ is also a prime). For instance, the prime $101$ is counted, when $m=6$, in column $N$ since $101+6=107$ is also a prime, but not counted in column $K$ since the next prime after $101$ is $103$ which is not distance $6$ from $101.$

\begin{array}{|c|c|c|} \hline m&N&K\\ \hline 2&1270&1270\\ 4&1264&1263\\ 6&\color{blue}{2538}&\color{blue}{2012}\\ 8&1303&801\\ 10&1682&953\\ 12&\color{blue}{2515}&\color{green}{1008}\\ 14&1546&513\\ 16&1275&354\\ 18&\color{blue}{2569}&\color{green}{537}\\ 20&1701&249\\ 22&1403&235\\ 24&\color{blue}{2578}&\color{green}{222}\\ 26&1402&91\\ 28&1519&102\\ 30&\color{red}{3451}&\color{green}{154}\\ 32&1246&35\\ 34&1357&36\\ 36&\color{blue}{2561}&\color{green}{55}\\ \hline \end{array}

It seems interesting that $N$ is about the same for all $m$ that are multiples of $6$, up to $m=36$, except for a spike at $m=30$ (this may have something to do with $30$ being divisible by the small prime $5$ (in addition to $2$ and $3$) whereas $6,12,18,24,36$ each is divisible only by primes $2,3$). I feel that a form of the pigeon-hole principle is relevant, if you put this many primes into this little space, some distances between these primes ought to repeat.

Mirko
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    The spike at $m=30$ is quite interesting. – Tito Piezas III Apr 20 '18 at 05:44
  • Thank you! By the way, it's probably nothing, but could you help me to check how many of 10 pairs were also twin(2) pairs? –  Apr 20 '18 at 05:50
  • @JC How do you define a 10 pair that is also a twin pair? I counted for how many of the first 10000 primes $p$ we have that $p+2$ as well as $p+10$ are also primes, the result is just a single occurrence, 3,5,13. If you count how many $p$ such that $p+2$ and $p+12=p+2+10$ are also primes, the result is $401$ many such triples (same for primes $p$ with $p-2$ and $p+10$ primes). If you count how many $p$ such that $p+10$ and $p+12=p+10+2$ are also primes, the result is $394$ such triples. You may try to verify these results on your own with a computer,just in case I didn't use the correct code – Mirko Apr 20 '18 at 06:07
  • @Mirko find all the 10 pair and store them in a list (Set 10) and then find all the 2 pair and store them in a list(Set 2), check the intersection. (i.e. in the table above 61,71,73 where 71 was 10 pair and a twin pair, so did 103 107...etc) –  Apr 20 '18 at 06:11
  • @JC 61,71,73 was counted in my previous comment, as p=61, p+10=71, p+12=73 (there were 394 such triples). How do you define a 10 pair, if it is indeed a pair, like (61,71), and you intersect with two-pairs, like (71,73) then the intersection is empty (since a 10 pair cannot also be a two-pair). So, you probably think of a prime $p$ such that some of the following are also primes p-2, p+2, p-10, p+10. There may be some variations how to count these (and I counted a couple of these possible variations in a previous comment). – Mirko Apr 20 '18 at 06:23
  • @Mirko Sorry, it's just to say the elements belonged to a 10 pair and a 2 pair, it's easier to check with list because equations sometimes makes mistakes. But if it's 394 then about 30% of 2 pairs intersect with about 23% of 10 pairs before 10000. (Difference about 7.5%, did you include p+12=p+10+2?) Still the portion was comparable large then it seemed. –  Apr 20 '18 at 06:34
  • @JC Whether lists are easier depends on language/software you would be using. I counted primes p, among the first 10000 primes, such that at least one of the following occurs: (a) p+2 and p+10 are primes, or (b) p-2 and p+10 are primes, or (c) p+2 and p-10 are primes, or (d) p-2 and p-10 are primes. The total of this is 797. Whether this is a lot is not clear, if you check a much larger range the result may not be convincing. If you do the same with the first 100000 primes the result is 5225. This does not prove the infinitude of twin primes. With 1000000 primes the result is 36566, go to bed – Mirko Apr 20 '18 at 06:54
  • @TitoPiezasIII For $m=60$, $N=3400$, and $K=2$. For $m=120$, $N=3397$, and $K=0$. (Both results computed for the first $10000$ primes.) – Mirko Apr 20 '18 at 07:02
  • @TitoPiezasIII : on "the spike at $m=30$ is quite interesting" - do you remember my older answer on a similar question https://math.stackexchange.com/a/1698703/1714 ? It gives a lot of insight about spikes of frequency... – Gottfried Helms Apr 20 '18 at 08:58
  • @JC : well, I've downloaded a file containing 50 000 000 primegaps provided by, I think, Chris Caldwell at his prime-pages. A nice resource for frequency analysis. If you don't find it there I can put this also on my homepage temporarily. – Gottfried Helms Apr 20 '18 at 13:24
  • @GottfriedHelms The number here was not "constructive". Sorry, the previous one was a typo, rewroted the program. I'd like to have the link.(Also, I'd like the prime number to be in a fixed range, <=1000,<=10000because I was thinking recently that the existence of primes had nothing to do with division and could be simply counted.) –  Apr 20 '18 at 13:28
  • for twin pairs and 10 pairs of first 10000 primes given by Mirko's calculation was 1264/1682=0.751486326 where as for primes <=1000000 8169/10934=0.74711907810499 . The rate was indeed reducing but at high portion around 0.75. As there was infinite twin primes, maybe it's also reasonable to guess that there was infinity 10 prime pairs. And the portion of those number was certainly very interesting...(Got to go to class) –  Apr 20 '18 at 13:39
  • <1000000 5349 elements both in twin pair and 10 pairs.5349/10934 =0.489207975 as 5349/8169 =0.6547925 –  Apr 20 '18 at 13:49
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    @JC - see this index-page for instance : http://primes.utm.edu/lists/small/millions/ This is 600 000 000 bytes in 50 files, and each record has ten numerical digits and 2 bytes for record-separation (12 bytes altogether times 50 000 000 primes= 600 000 000 bytes) – Gottfried Helms Apr 20 '18 at 21:14