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Trying to avoid shelling out hundreds of dollars so I'm using what I can for free online. This is what I've come up with so far:

https://dl.dropboxusercontent.com/u/76769933/TwinPrimes%203Podd.cdf

Trying to use this to demonstrate a possible way to prove the twin prime conjecture:

For every prime $p$ greater than 2, there exists one or more twin primes as follows: $(3pn-4, 3pn-2)$ where $n$ is some positive odd less than or equal to $p$.

With the free version, I can only go so far.

My questions are these:

  1. Would someone help test for a counter-example. Maybe refine this code to only print if a counter-example exists in the first million or so primes.
  2. Would someone perhaps plot Length[twin] vs. $p$ to show that as $p$ increases, so does the number of twins found.

I realize this question requires simultaneously a lot of knowledge of primes and Mathematica, as well as requiring you to look through code to fully understand the question, so thanks in advance.

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  • $\begingroup$ Just messed with this some. Found this: as $p$ increases, so does the number of twin primes. I've also found that it is highly unlikely a counter-example will be found. The highest % through odds before a twin prime is found is 78%; the next highest being only 40%, with the average percent dropping heavily as $p$ increases. The proof of the twin prime conjecture is right here in the math; a nugget waiting to be dug up by someone who simply understands what's going on. $\endgroup$ – Elem-Teach-w-Bach-n-Math-Ed Mar 28 '16 at 4:35
  • $\begingroup$ If this result is original, it is extremely impressive and paper-worthy. I do wonder, however, if it's an already known result. In either case, I suggest keeping this question open. $\endgroup$ – user1722 Mar 28 '16 at 6:05
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    $\begingroup$ I get worried when I see phrases such as "one or more unique xxx". It's just not clear what that means. $\endgroup$ – Daniel Lichtblau Mar 29 '16 at 14:42
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    $\begingroup$ @DanielLichtblau Which is the unclear part? "One or more" or "unique"? Let me know and I'll clarify. $\endgroup$ – Elem-Teach-w-Bach-n-Math-Ed Mar 29 '16 at 18:26
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    $\begingroup$ (3) I do not much like the subject header. This is not a proof of anything. $\endgroup$ – Daniel Lichtblau Mar 30 '16 at 2:41
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No counterexample.

ClearAll[p]
p = 2;
primes = Reap[While[p <= 1000000, Sow[p = NextPrime[p]]]][[2, 1]];

Do[
  Table[
   If[PrimeQ[3*primes[[i]]*j - 4 (* or 2 *)],
    Continue[]
    ]
   ,
   {j, 1, primes[[i]], 2}
   ];
  Return[i]
  ,
  {i, 1, Length@primes}
  ] // AbsoluteTiming

{76.1693, Null}

twinPrimes = Table[
   {
    primes[[i]],
    Pick[#, PrimeQ@#, True] &@(3*primes[[i]]*Cases[Range@primes[[i]], _?OddQ] - 4),
    Pick[#, PrimeQ@#, True] &@(3*primes[[i]]*Cases[Range@primes[[i]], _?OddQ] - 2)
    },
   {i, 10}
   ];

twinPrimes // MatrixForm

enter image description here

ListLinePlot[
 {
  {#[[1]], Length@#[[2]]} & /@ twinPrimes,
  {#[[1]], Length@#[[3]]} & /@ twinPrimes
  },
 AxesLabel -> {"prime number", "twins amount"},
 ImageSize -> Large
 ]

enter image description here

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    $\begingroup$ Technically, aren't these the primes under one million and not the first one million primes? $\endgroup$ – user1722 Mar 28 '16 at 5:15
  • $\begingroup$ @barrycarter You are correct. These are primes under 1M (around 79K primes). Maybe later I will check more primes. $\endgroup$ – Alexey Golyshev Mar 28 '16 at 5:44
  • $\begingroup$ True, but still, awesome work! Thanks for not ignoring a question on hold! As I commented on the original question, the indication is strong that no counter-example will be found. Maybe a graphic plot like my 2nd question would help show this? $\endgroup$ – Elem-Teach-w-Bach-n-Math-Ed Mar 28 '16 at 5:51
  • $\begingroup$ @user3363795 Updated. Plot of the first 2K twinPrimes. $\endgroup$ – Alexey Golyshev Mar 28 '16 at 6:10
  • $\begingroup$ That plot is sick! I knew this was convincing evidence of the twin prime conjecture, but that is awesome! It appears nearly linear! $\endgroup$ – Elem-Teach-w-Bach-n-Math-Ed Mar 29 '16 at 0:43
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Ok, so I found I can go a little further on lab.wolframcloud.com/app. Here's what I've done with a lot of ideas from @Alexey Golyshev.

[Updated 3/29 18:21] Download and run this notebook: https://dl.dropboxusercontent.com/u/76769933/Twinprimeplotting.nb

to get this: ![enter image description here

Don't know how much further I can go with this. Need someone to stand on my shoulders and take this further!

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  • $\begingroup$ Added best fit line and parabola. That parabola is so close to a line, there must be something going on here. I may post this on math.stackexchange just to see if someone can explain why this line holds so well. $\endgroup$ – Elem-Teach-w-Bach-n-Math-Ed Mar 30 '16 at 0:25
  • $\begingroup$ "so close to a line" - did you have a look at the coefficient of determination? $\endgroup$ – J. M.'s torpor♦ Mar 30 '16 at 1:59
  • $\begingroup$ This might be better asymptotically: In[43]:= curve = FindFit[data, x/(b*Log[x])^2, {b}, x]; new = x/(b*Log[x])^2 /. curve Out[44]= (5.45108297312 x)/Log[x]^2 $\endgroup$ – Daniel Lichtblau Mar 30 '16 at 2:10
  • $\begingroup$ To understand where that suggested fit arose, take a series at infinity of the integral estimate I put in a comment under the original post. $\endgroup$ – Daniel Lichtblau Mar 30 '16 at 2:43
  • $\begingroup$ @J.M. Sorry, elem. math teacher with only a Bach. degree. Define "coefficient of determination", and how I would find it. $\endgroup$ – Elem-Teach-w-Bach-n-Math-Ed Mar 30 '16 at 5:31

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