An approach to the Twin Prime Conjecture

[galoisjr]

The prime numbers are the multiplicative building blocks of the integers. Even so, their distribution escapes all methods of rationalization. As with building a pyramid, the primes are most densely distributed near zero, the point of origin, and as we move towards larger numbers the primes are less dense. However, counterintuitive to this notion, there do exist arbitrarily large consecutive prime numbers which are separated by an interval of only 2. The Twin Prime Conjecture asserts that these pairs form an infinite set.

The proof of this conjecture has eluded number theorists for quite sometime. Being only in my junior year of my undergraduate education, and never having taken a course in number theory, I do not think that I have the tools to rigorously proof the conjecture. But I do think that I have found a path towards a constructive proof.

I sort of stumbled into this while looking at a list of factorials, but I realized that for quite a few integer j, a ball of radius 1 about j! contains either two primes, or a prime and a square. After toying with this notion for awhile, I made a conjecture of my own which I am fairly certain of:

For each n in the integers there exists some k in the integers such that (k*n!-1, k*n!+1) is a twin prime pair.

I have no idea where to begin a proof of the validity or invalidity of this statement. And this statement alone is not sufficient proof of the Twin Prime Conjecture since there would be some repetition of the twin prime pairs as n increases and we cycle back through the integers to find a suitable k. However, if the statement is true, then finding a bound on k will be sufficient proof of the TPC.

Any comments, skepticism, ideas, etc. are highly encouraged.

Thanks in advance

 

[jedishrfu]

i'd test it by some computer runs first. Its an advantage we have that earlier generations didn't have.

and I'd talk to a prof because you never know it could be the corner stone of a cool proof and you've just given it away.

If you could find a list of twin primes you could test your formula on it and if it bears out then work it the other way to find some new twin primes then move on to a proof. prime testing via computer can take a lot of time and cpu resources so you might want to get an account on a supercomputer near you. It'd be an interesting research project, if nothing you'd learn how to do math research via programming.

 

[dodo]

I think your statement would constitute a proof of the twin prime conjecture, regardless of the repetitions you mention. If your statement is true, for any n you choose there exists a k such that (k*n!-1, k*n!+1) are twin primes. Now choose another n > k*(previous n)!+1; there will be another pair of twin primes, higher (thus distinct) than the previous. And so on; since there are infinite n's to choose from in this manner, there will be infinite pairs. Note that nobody requires the proof to account for each and all pair of twin primes, just to prove that there is an infinitude of them.

The problem could be that your statement may turn out to be just as difficult to prove as the conjecture itself. You mention that you're fairly certain of it, yet have no idea on how to prove it; it is nice to be in love with an idea, but that doesn't take you any closer to a proof. I gather most mathematicians would have a similar feeling about the conjecture itself, and put it aside until a hint of a proof comes along. So, try to think of reasons why your statement may be true (reasons that you would not apply to the conjecture itself, such as numerical evidence).

 

[fleem]

Beyond the initial definition of prime numbers, they effectively have a random component (so far) in our eyes. That random component prevents us from proving anything else about them except statistical behavior, and disproofs of conjectures by finding counter examples. The twin prime conjecture cannot be proven by counter example, so it is not yet provable. If ever someone is able to find a formula predicting prime numbers (with algorithmic complexity that is not related to the magnitude of that number), then they will no longer be "effectively random" in our eyes and other proofs will be possible.

 

[DivisionByZro]

How do you propose one could find such a "k"? I was going to write a program to calculate a few values, but there really is no set restriction on "k", so a program might not be feasible right now.

 

[epsi00]

[q]and I'd talk to a prof because you never know it could be the corner stone of a cool proof and you've just given it away.[/q]

his post has a unique time stamp and has been posted on a public forum. it's easy to prove who posted it if it ever come to that. I would say it's way easier to prove than the twin prime conjecture itself.
I think it is a fresh idea that needs to be studied seriously and like it was mentioned before, numerical experiments will quickly tell if the OP is on the right track.

 

[dodo]

[...] numerical experiments will quickly tell if the OP is on the right track.

Not sure how these experiments could be. My impression is that his hunch is based on a few, very small values of n! because, as n gets large, it is not that obvious that the neighbors of n! are likely to be prime. Here are a few more (and still too few) values of n!, with n!-1 and n!+1 factorized; notice the values you get from 15 onward.

 n                  n!   factors of n!-1         factors of n!+1
                       |                       |
 3                   6 | 5                     | 7
 4                  24 | 23                    | 5 5
 5                 120 | 7 17                  | 11 11
 6                 720 | 719                   | 7 103
 7                5040 | 5039                  | 71 71
 8               40320 | 23 1753               | 61 661
 9              362880 | 11 11 2999            | 19 71 269
10             3628800 | 29 125131             | 11 329891
11            39916800 | 13 17 23 7853         | 39916801
12           479001600 | 479001599             | 13 13 2834329
13          6227020800 | 1733 3593203          | 83 75024347
14         87178291200 | 87178291199           | 23 3790360487
15       1307674368000 | 17 31 31 53 1510259   | 59 479 46271341
16      20922789888000 | 3041 6880233439       | 17 61 137 139 1059511
17     355687428096000 | 19 73 256443711677    | 661 537913 1000357
18    6402373705728000 | 59 226663 478749547   | 19 23 29 61 67 123610951
19  121645100408832000 | 653 2383907 78143369  | 71 1713311273363831
20 2432902008176640000 | 124769 19499250680671 | 20639383 117876683047

Edit:
Used Maxima to get a few more. The 'primes' listed below are, possibly, "prime with high probability", as Maxima may be using the Miller-Rabin test to determine when to try to factorize further; I just don't know.
Under this caveat, only the following are 'primes': 27!+1, 30!-1, 32!-1, 33!-1, 37!+1 and 38!-1. The rest are clearly composite.

 n                                               n!   factors of n!-1                                       factors of n!+1
                                                    |                                                     |
21                             51090942171709440000 | 23 89 5171 4826713612027                            | 43 439429 2703875815783
22                           1124000727777607680000 | 109 60656047 170006681813                           | 23 521 93799610095769647
23                          25852016738884976640000 | 51871 498390560021687969                            | 47 47 79 148139754736864591
24                         620448401733239439360000 | 625793187653 991459181683                           | 811 765041185860961084291
25                       15511210043330985984000000 | 149 907 114776274341482621993                       | 401 38681321803817920159601
26                      403291461126605635584000000 | 20431 19739193437746837432529                       | 1697 237649652991517758152033
27                    10888869450418352160768000000 | 29 375478256910977660716137931                      | 10888869450418352160768000001
28                   304888344611713860501504000000 | 239 156967 7798078091 1042190196053                 | 29 10513391193507374500051862069
29                  8841761993739701954543616000000 | 31 59 311 26156201 594278556271609021               | 14557 218568437 2778942057555023489
30                265252859812191058636308480000000 | 265252859812191058636308479999999                   | 31 12421 82561 1080941 7719068319927551
31               8222838654177922817725562880000000 | 787 992078233 10531763920894209415469               | 257 95101 3038779 110714485281307653167
32             263130836933693530167218012160000000 | 263130836933693530167218012159999999                | 2281 652931 61146083 2889419049474073777
33            8683317618811886495518194401280000000 | 8683317618811886495518194401279999999               | 67 50989 175433 143446529 101002716748738111
34          295232799039604140847618609643520000000 | 10398560889846739639 28391697867333973241           | 67411 4379593820587205958191075783529691
35        10333147966386144929666651337523200000000 | 37 71 3933440413546305645095794190149676437         | 137 379 17839 340825649 32731815563800396289317
36       371993326789901217467999448150835200000000 | 155166770881 2397377509874128534536693708479        | 37 83 739 1483 165202043 669043459524628666916941
37     13763753091226345046315979581580902400000000 | 53 439 1477897 6154980127 65031782905798661084563   | 13763753091226345046315979581580902400000001
38    523022617466601111760007224100074291200000000 | 523022617466601111760007224100074291199999999       | 14029308060317546154181 37280713718589679646221
39  20397882081197443358640281739902897356800000000 | 41 10949 99563 456382297346497242065582795509270897 | 79 57554485363 146102648914939 30705821478100704367
40 815915283247897734345611269596115894272000000000 | 9190813196017748117 88775091588350692405196340547   | 41 59 277 217823 16558103 142410167827 2370686450613664429

 

[jedishrfu]

How do you propose one could find such a "k"? I was going to write a program to calculate a few values, but there really is no set restriction on "k", so a program might not be feasible right now.

K might be limited to n.

 

[galoisjr]

Thanks to everyone for the immediate responses. And especially to Dodo for computing the case of k = 1.

If I new of some formula to find such a k then the conjecture would be quite simple to prove. This is the reason why I posted this here, so that I could gain some insight because like I said, I have never taken a course in number theory and I attend a school which only offers an applied mathematics track. I wish now that I could have done pure but I am already a junior.

Any more insight into this problem that I do gain will be posted in this thread and hopefully we might be able to pool our knowledge to find a solution.

Do note that if we let n = 2 then my conjecture this is merely a restatement of the TPC.

 

[galoisjr]

K might be limited to n.

It truly does not matter what k is limited to just that it is limited. For instance if k was limited to n! would be sufficient. I however do not think that k could be limited to just n to produce such a result.

For proof of my conjecture, it does not matter what k is, just that it exists. I was thinking maybe modular arithmetic? I am fairly certain of the conjecture and it may even follow directly from closure of the integers under addition and multiplication and the fundamental theorem of arithmetic. I think if this statement is not true then it would imply that the integers are not closed, a fact which we know to be false.

 

[galoisjr]

Not sure how these experiments could be. My impression is that his hunch is based on a few, very small values of n! because, as n gets large, it is not that obvious that the neighbors of n! are likely to be prime. Here are a few more (and still too few) values of n!, with n!-1 and n!+1 factorized; notice the values you get from 15 onward.

 n                  n!   factors of n!-1         factors of n!+1
                       |                       |
 3                   6 | 5                     | 7
 4                  24 | 23                    | 5 5
 5                 120 | 7 17                  | 11 11
 6                 720 | 719                   | 7 103
 7                5040 | 5039                  | 71 71
 8               40320 | 23 1753               | 61 661
 9              362880 | 11 11 2999            | 19 71 269
10             3628800 | 29 125131             | 11 329891
11            39916800 | 13 17 23 7853         | 39916801
12           479001600 | 479001599             | 13 13 2834329
13          6227020800 | 1733 3593203          | 83 75024347
14         87178291200 | 87178291199           | 23 3790360487
15       1307674368000 | 17 31 31 53 1510259   | 59 479 46271341
16      20922789888000 | 3041 6880233439       | 17 61 137 139 1059511
17     355687428096000 | 19 73 256443711677    | 661 537913 1000357
18    6402373705728000 | 59 226663 478749547   | 19 23 29 61 67 123610951
19  121645100408832000 | 653 2383907 78143369  | 71 1713311273363831
20 2432902008176640000 | 124769 19499250680671 | 20639383 117876683047

Edit:
Used Maxima to get a few more. The 'primes' listed below are, possibly, "prime with high probability", as Maxima may be using the Miller-Rabin test to determine when to try to factorize further; I just don't know.
Under this caveat, only the following are 'primes': 27!+1, 30!-1, 32!-1, 33!-1, 37!+1 and 38!-1. The rest are clearly composite.

 n                                               n!   factors of n!-1                                       factors of n!+1
                                                    |                                                     |
21                             51090942171709440000 | 23 89 5171 4826713612027                            | 43 439429 2703875815783
22                           1124000727777607680000 | 109 60656047 170006681813                           | 23 521 93799610095769647
23                          25852016738884976640000 | 51871 498390560021687969                            | 47 47 79 148139754736864591
24                         620448401733239439360000 | 625793187653 991459181683                           | 811 765041185860961084291
25                       15511210043330985984000000 | 149 907 114776274341482621993                       | 401 38681321803817920159601
26                      403291461126605635584000000 | 20431 19739193437746837432529                       | 1697 237649652991517758152033
27                    10888869450418352160768000000 | 29 375478256910977660716137931                      | 10888869450418352160768000001
28                   304888344611713860501504000000 | 239 156967 7798078091 1042190196053                 | 29 10513391193507374500051862069
29                  8841761993739701954543616000000 | 31 59 311 26156201 594278556271609021               | 14557 218568437 2778942057555023489
30                265252859812191058636308480000000 | 265252859812191058636308479999999                   | 31 12421 82561 1080941 7719068319927551
31               8222838654177922817725562880000000 | 787 992078233 10531763920894209415469               | 257 95101 3038779 110714485281307653167
32             263130836933693530167218012160000000 | 263130836933693530167218012159999999                | 2281 652931 61146083 2889419049474073777
33            8683317618811886495518194401280000000 | 8683317618811886495518194401279999999               | 67 50989 175433 143446529 101002716748738111
34          295232799039604140847618609643520000000 | 10398560889846739639 28391697867333973241           | 67411 4379593820587205958191075783529691
35        10333147966386144929666651337523200000000 | 37 71 3933440413546305645095794190149676437         | 137 379 17839 340825649 32731815563800396289317
36       371993326789901217467999448150835200000000 | 155166770881 2397377509874128534536693708479        | 37 83 739 1483 165202043 669043459524628666916941
37     13763753091226345046315979581580902400000000 | 53 439 1477897 6154980127 65031782905798661084563   | 13763753091226345046315979581580902400000001
38    523022617466601111760007224100074291200000000 | 523022617466601111760007224100074291199999999       | 14029308060317546154181 37280713718589679646221
39  20397882081197443358640281739902897356800000000 | 41 10949 99563 456382297346497242065582795509270897 | 79 57554485363 146102648914939 30705821478100704367
40 815915283247897734345611269596115894272000000000 | 9190813196017748117 88775091588350692405196340547   | 41 59 277 217823 16558103 142410167827 2370686450613664429

I also noticed this, however I was computing them by hand. But this was my reason for introducing k.

 

[dodo]

I also noticed this, however I was computing them by hand. But this was my reason for introducing k.

Ok. Hope the following helps.
I used Maxima to produce an output like

                                     n = 3
                                     k = 1
                               primep(5) = true
                               primep(7) = true
                                     k = 2
                               primep(11) = true
                               primep(13) = true
                                     ...

and then some text processing to produce the 'graphic' below.

In the graphic, there is one line for each value of n, from 3 to 50; and each line has pairs of characters, like ".." or ".|", one pair for each value of k (from 1 to 100). The pair of characters on each column represent the primality of k*n!-1 and k*n!+1, respectively: a dot for "not prime", a vertical bar for "prime". Whenever both happen to be prime (twins), instead of "||" two asterisks "**" are printed, to make them stand out.

Likely you'll need larger and larger k's to produce a result... but the point was: as primes get scarcer and scarcer, why should there be a twin prime pair for each n?

For instance if k was limited to n! would be sufficient.

Why n! ? (I assume you mean "a limit for the purpose of gathering evidence with the computer", since in your actual conjecture k should be left unlimited, of course.)

I still wonder why do you feel so certain of your conjecture... and why in more degree than, say, feeling certain "a priori" of the twin prime conjecture itself.

Do note that if we let n = 2 then my conjecture this is merely a restatement of the TPC.

I got lost on this one. Don't know what you mean.

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[galoisjr]

I was just saying that if it could be shown that there exists some k in the set of integers bounded above by n! such that my conjecture holds then it would be a sufficient proof of the TPC. What I was trying to say is that k is arbitrary in my conjecture but it needs some bound to be proof of the TPC. Someone suggested bounding k by n. However I don't think that in that case my conjecture would hold. I don't know if it would hold if it was bounded by n!, it was just a suggestion without any evidence to back it up.

When I said that if n=2 then it is a restatement of the TPC is actually not true. What I was getting at was that when n= 2 if there exists a k such that (2k-1, 2k+1) is a twin prime pair.

This is actually a restatement of the TPC only if there exists infinitely many k such that (2k-1, 2k+1) is a twin prime pair.

I honestly have no idea why I feel so strongly about the conjecture, I scribbled a few things on a sheet of paper about a year ago which I am just looking at now, and I am trying to figure out what exactly I meant by them.

Anyway, thanks for all of your help though Dodo.

 

[galoisjr]

I just found the second sheet of paper that seems to convey my reasoning for why i think that the twin prime conjecture must be true.

Traditionally when thinking about the density of the primes, we use the pi(x) function. However, this function gives densities in blocks of numbers of an increasing size which is the reason why the distribution of the primes and connections between the primes are hard to judge. It's somewhat analogous to someone who has never done any arithmetic trying to add fractions with different denominators for the first time.

But we can simplify the problem. Instead of trying to determine how the primes are distributed throughout the entire set of numbers, we could instead consider consecutive intervals of a constant magnitude. Say, for instance, how primes are distributed in the 1st block of of a hundred integers, second block, and so on.

These measurements generate a step function which judging from the growth of pi(x), decreases on average from the 1st to the kth interval. Thus we can see that the step function can be approximated by a probability density function, specifically a normal curve. This simplifies the problem since there are vast repertoires of knowledge about normal distributions. It also tells us that the primes and their additive inverses are approximately normally distributed about the zero.

Since we are looking solely at the density of primes in the kth consecutive interval, we can conclude that the error between the normal curve and the step function in the kth interval should be the dependent only on the number of primes in that interval and thus the gaps between the primes in that interval: error(k) = N(x in k) - step(k) + C where C is a constant determined by the error in the previous k-1 intervals and should then be the sum of the errors in the previous k-1 intervals.

Now, we also know that there are an infinitude of primes. The error is dependent on the number of primes in the kth interval. We know that the pi function does not stop increasing. Thus our step function oscillates about 1. Since it only contains integer values, while there must be intervals that contain no primes, there must also be intervals which contain two primes, in fact an infinite number of such intervals. Since the gap between these primes is seemingly random in these intervals and must be between 2 and some finite number n, we deduce that since there exists a number, smaller than the number of intervals yet still infinite, of occurrences of each prime gap between 2 and n. In other words, since the prime gap occurs once, it is reasonable that it occurs an infinite number of times.

 

[dodo]

I am in a similar situation as you; while I did take some pregrad courses in pure math (number theory among them), what you're exploring enters analytic number theory, which is a postgrad study; so here is where I stop talking. :)

One thing I would ask is to clear up your reasoning. I though the step(k) function was counting the number of primes in the kth interval (say, how many primes are there between 100k-99 and 100k), and I can't follow how do you reach conclusions about the actual gaps between primes. And again, the argument is about the TPC itself and not about your conjecture - which was interesting, but somewhat unsupported.

But hey, keep banging it.