I am an amateur mathematician who undertook many years ago to try and prove the classic "Twin Prime Conjecture". Simply put the conjecture is as follows: Are there an infinite number of primes which are two apart? Recall that a prime number is a number divisible only by itself or one. Some example of prime twins are 3,5 and 17,19. "Who cares?!" asks the typically truculent student. "I won't need prime twins when I'm doing my future job" s/he typically continues. Other than the fact that the conjecture is very hard to prove, mathematicians themselves have a difficult time explaining what is so compelling about the problem.
Underpinning the problem is the fact that primes appear to behave randomly. That is they tend to pop up like groundhogs every so often on the number line landscape with both decreasing probability and predictability. As a result it is very hard to say anything reliable about primes other than that we know them to be infinite thanks to a famous proof by Euclid. As a result of the lack of predictable things to say about primes, mathematicians lack the building blocks to build up a proof about the infinity of prime twins.
The closest we've gotten is an attempt by mathematicians Goldston, Pintz and Yildirim. Loosely put, they were able to show that there is an infinite supply of primes with a 'small gap'. Small here means greater than 2 (meaning not prime twins) but other small gap prime constellations. Many hold out hope that they will eventually be able to pour in enough ingredients to their proof to ratchet down the gap to 2 and thus finally prove the prime twin conjecture. Many again also think this will solve the long sought prime twin conjecture proof and all of its problems. I'm not one of these people.
I apologize in advance for what I'm about to say will sound arrogant, but I find the work of Goldston, Pintz and Yildirim to be largely wasted. Yes, I realize I have just attacked the newest Gods of mathematics, but their proof isn't answering the real question. Before I continue, realize I'm not attacking the intellect of the aforementioned, I find them to be geniouses in the truest sense of the word. My curt tone should be construed as my strongest possible suggestion to them to refocus their efforts towards the more global problem as follows: The Prime Twin Conjecture isn't enigmatic because there's something special about prime twins, it's enigmatic because there's something special about primes in general. The real question behind the twin prime conjecture is: "are all alloweable prime constellations infinite?"
A prime constellation is a group of primes. A prime twin is one type of prime constellation. Another is p, p+2, p+6, such as 11, 13, 17. Not all constellations are allowable. The constellation p, p+2, p+4 is not allowable (except in the case of 3,5,7) because there is always a multiple of 3 in this constellation so one of the entries is guaranteed not to be prime. In order to answer this question about all allowable prime constellations being infinite, you need to look at the randomness of primes, which is something mathematicians are fearful of doing. The murky depths of randomness ask mathematicians to navigate in right brain waters which is something they loathe doing. For this reason, mathematics is a self limiting study.
The work of so many mathematicians seeking the holy grail of the prime twin conjecture proof is akin to the efforts of early American space scientists in seeking to create a pen which would operate effectively in space. They devised all manner of engineering skills and came up with an elaborate 'astronaut pen'. The Russian scientists faced with the same problem, chose not just to look at the problem at hand, but at the intent of the posed problem. The underlying need was for a writing device that operated in zero gravity. By looking at the underlying question they solved the problem, with a pencil.
This is a good allegory which demonstrates the problems of current attempts to solve the twin prime conjecture. Mathematicians are employing all manners of existing techniques and skills in order to solve the problem without looking at the larger problem of the allowable constellations. If any of them succeed they'll be wildly celebrated as mathematic heroes for reasons uncertain to this author. Yes, certainly an accolade or two is due, but they will have missed the entire point of the question. The second a twin prime conjecture proof is proffered, this author will immediately ask, "well what about all the other constellations?" I've seen the approach used by most mathematicians and can assure the lay reader that the methods and the proof of the twin prime conjecture will likely never answer the larger question of the infinitude of all allowable constellations.
In order to solve the problem of the infinitude of the constellations, we must look at the characteristics of the primes -- namely randomness. There are huge problems with randomness in mathematics. "It is evident that the primes are randomly distributed, we just don't know what 'random' means." -- R.C. Vaughan Random is one of those better left to the right side of the brain type concepts that mathematicians aren't suited to deal with. However, this author offers a solution: "Determinism cross Recursive Self Complication equals randomness." -- M. C. Winer That is a deterministic process with some level of complexity can recursively take that complexity and add more complexity to itself. If we accept that primes are random according to this definition, then it follows, after some study, that all allowable prime constellations are infinite. This is the direction mathematics must follow.
Anyone interested in reading more is welcome to visit: http://www.rankyouragent.com/primes/primes.htm which proves that all allowable prime constellations, including prime twins, are infinite. The observant may have noted that I just claimed to have proved the Twin Prime Conjecture. The observant would have observed correctly, moreover, I proved it many years ago. It's not hubris; it's just a fact that anything that can so vehemently be denied as rubbish can but only be correct. This method of proof has been discussed to death on math Usenet groups where no good idea goes unpunished. Be not afraid, if you really want to understand why the primes behave the way they do and understand the deeper meanings of the primes, then lend some patience and some time to the proof above. If you just want to win a prize and solve the most banal manifestation of a miraculous phenomenon, please turn your attention to the twin prime conjecture. Winning a large cash prize for solving the simplest question of a complicated concept is "nice work if you can get it". (The Clay Institute offers $1 million to anyone who can settle the Riemann Hypothesis which is closely tied to the twin prime conjecture.) Understanding and, worse, trying to explain the deeper implications is a thankless task, but well worth it personally. |
