Can the twin primes conjecture be proven using probability?
If I understand what you’re saying, then No: no amount of probability is accepted as a mathematical proof.
Oh cause I thought if you assume that
2; 4; 6; 8; 10; 12
were the only prime differences in the world then the chances that the next prime difference is 2 would be 1:6. If the set of differences is extended to 24 then the chance that the next difference is 2 would be 1:12.
Therefore the chance that the next difference is 2 can be written as 1:n where n is the number of potential differences. As a result unless n is infinity then the probability will always be above zero. Therefore there are an infinite number of twin primes.
This is assuming that primes are random and have a trend of increasing differences.
Short answer: no.
Longer answer:
Most number theory theorems are absolute statements. Meaningful proofs would have to make an absolute statement about the existence of an infinitude of twin primes (or prove that there are only a finite number of them).
That’s not to say that statements about the density of twin primes would not be useful. Anything we can determine would be a valid contribution to theory. But that would not constitute a proof (or disproof) of the conjecture itself.
Cause I developed a theorem where all primes differences could be categorised into three groups. When you tally the number of differences of each group the frequency of all three groups will equal from time to time
There are some significant problems with this line of reasoning.
One is how you are defining probability. In this case, it appears you are defining your probabilistic elements in terms of differences between successive primes. But there are infinitely many primes, so how can you define a probability of a particular difference in success primes at all? There are potentially infinitely many cases for each potential difference.
Another problem is that you can have an arbitrarily large difference between primes. The difference between primes is NOT limited to a specific, finite range.
Another problem is that primes aren’t random. Probability theory relies on a notion of randomness. In this case, there is NO randomness. You can think of the sequence of differences between successive primes as pseudo-random, but that’s not the same thing as random at all. In other words, if you have a prime number, the next prime number (i.e. the difference between it and the previous one) isn’t random at all. It’s deterministic. It only seems random because we often treat “unknown” as “random” when that’s not the case at all.
As noted above, prime differences are NOT random. They form, at best, a pseudo-random sequence. Treating them as a probabilistic group is already questionable. It may be useful to think of them that way, but it gives no real, provable insight into the distribution of primes.
There are hundreds (more technically, gazillions) of number sequences that are known to stop. Therefore no probability of their continuing based on the known elements could possibly be true. We cannot use the elements of the sequence of known twin primes to determine the probability of their continuing.
A theorem is only a theorem if it has been proved. You can’t prove this so what you have is at best a hypothesis or conjecture. You could try posting your work here and let the professional mathematicians tear it apart, but I’ll bet sight unseen that it doesn’t even resemble the proof of a theorem.
I know that sequence I wrote previously was just an example. But if one were to count the number of time the differences
2;4;6;8;10;12
Occurred as you tend to infinity they would occur the same of times. That would also hold true for any sequence of differences you cared to write down.
When you say pseudo-random what do you mean?
But the point is knowing whether the differences have a specified property. Knowing that the differences exist says nothing about that property.
Do you honestly think that the entirety of professional mathematicians who have been working on this problem since 1849 have missed such an elemental proof? Do you know anything about the actual math that constitutes the body of work that has been whittling down the gap to 5414?
Making statements about math is not math. Only math is math.
Really? You can prove this? Or is it something you are assuming because it sounds like it makes sense.
If it’s something you can prove, that would actually be a major development. But if it’s just something that appears to hold for the few numbers you checked, it’s not as useful.
Also, how big are the differences you are checking? As I noted above the differences between primes can be arbitrarily large. They aren’t limited to such small numbers.
Something that looks random but isn’t actually random. Difference between successive primes aren’t actually random. They’re deterministic. Given a prime, you can always find the next prime difference, and it never changes. That’s the opposite of random.
And if they aren’t random, you can’t really treat them as random and expect to get valid results.
You may be interested in this thread—Mathematical theorems that are seemingly true but are then violated by a VERY large number?—or in the article which is linked to in its original post, on recent progress in the twin primes conjecture.
I can’t prove it unfortunately. By your definition of pseudo-random would it be possible to create a formula that would generate primes?
I can’t prove that these three cases of prime differences all have the same frequency up to infinity as I am only in matric. But I can send you an e-mail to describe them if you would like to have a look at them because maybe if you prove the frequency of each case is the same as you tend to infinity that would help solve twin primes.
If we put the OP’s statement into more rigorous terms, I think it would go something like this:
“Given any two positive even numbers e[sub]1[/sub] and e[sub]2[/sub], and given the set of prime intervals up to some number n, as n tends to infinity, the ratio of the proportion of intervals which is equal to e[sub]1[/sub] to the proportion of intervals which is equal to e[sub]2[/sub] tends to 1”.
Note that, although this statement can be more succinctly summarized in terms of probability, it does not actually require any probability in its statement. And if this statement were true, I’m pretty sure that, together with other theorems known about prime numbers, it would imply the truth of the Twin Prime Conjecture, through a fairly straightforward demonstration.
The kicker, of course, would be proving that statement true. Even if it is true (which I don’t know), it would be quite difficult to prove it.
It depends on what you mean by “formula”.
If you mean a polynomial that only spits out prime numbers, then no. But if you mean some sort of procedure which generates only primes, sure. Computers can be programmed to do this. And because primes don’t occur randomly, the outputs of different programs will always generate the same output because primes are fixed, not random. These aren’t always going to be elegant or simply stated, though.
A not wholly dissimilar analogue is the decimal expansion of pi. The n-th digit of pi is fixed. It’s deterministic. It’s not random at all. But, in another sense, we can treat the digits of pi as almost-kinda-sorta random. If you were to pick one of the first several million digits of pi at random, it could have any value from 0-9 with roughly even likelihood.
But that’s not the same as actually being random. The 58317th digit of pi can only have one value, just as the 58318th digit can only have one value. The digits are not actually random, even if you can kind of treat them that way in certain respects.
Note we don’t have a simple function to generate these digits. To get them, we have to perform lengthy calculations. The exception is in hexadecimal, where there actually is a short, simple formula for generating the n-th digit of the hexadecimal expansion of pi.
So, even if something is not random, it doesn’t always follow that there’s a simple function for producing it. Sometimes there is (like the hex expansion of pi) but there often is not (the decimal expansion of pi).
I’m going to stop you there. There are, in fact, infinitely many achievable prime differences.
Proof: Let N be an arbitrarily large value, and consider N! + 2, N! + 3, …, N! + N. These are N - 1 many values in a row, all of which are composite (N! + 2 is divisible by 2, N! + 3 is divisible by 3, etc.). Thus, there is a prime difference (between the largest prime below N! + 2 and the smallest prime above N! + N) of size at least N. As N was arbitrarily large, this means there are arbitrarily large prime differences, which amounts to the same thing as there being infinitely many achievable prime differences. Q.E.D.
There are other difficulties with your argument, as others have noted, but I figured, this should be pointed out as well.
(I see now that Antibob already pointed this out above. And Chronos’s reformulation could still work (er, well, if it could work, you know?..))
Speaking of which:
What other theorems do you have in mind?
I know how that N! function works and it doesn’t describe the number of potential differences accurately at all. The number of differences has constraints and the way each prime can be in relation to the 6n±1 is limited and defined. I agree that my argument falls apart but not for that reason. Differences cannot repeat consecutively unless the difference is a multiple of 6.
I suspect that the Prime Number Theorem alone would be enough, though I haven’t calculated it out. Failing that, aren’t there theorems to the effect that there are an infinite number of prime pairs separated by no more than N, where N is some large number? That would certainly be enough.
And Great Antibob, one can easily use the formula for digits of pi in hexidecimal to construct a similar formula for digits of pi in any base which is a power of 2. It’s also possible that there might be a formula for other bases, though nobody yet knows of any (I would have guessed it to be unlikely, but then I would have guessed the same thing of the hexidecimal formula, before it was discovered).