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Another Breakthrough in Prime Number Theory
Battal Boy writes "From aimath.org: Dan Goldston and his Turkish colleague Yalcin Cem Yildirim have smashed all previous records on the size of small gaps between prime numbers. This work is a major step toward the centuries-old problem of showing that there are infinitely many 'twin primes': prime numbers which differ by 2, such as 11 and 13, 17 and 19, 29 and 31,...I am especially proud of this achievement as Yalcin is a close friend of mine from way back! You may also want to check out the Mercury News Article and Dan Goldston's home page where you can see a photo of Dan's back being slowly but surely broken by two of his children ..." Finding patterns in primes seems to be all the rage.
One reason that it is intuitivly possible that there are an infinite number of twin primes is that it is possible to generate numbers that are relativily prime. For example, multiply the first three primes: 2,3, and 5. You get 30. Add or subtract one from 30 and you will get a number that is relatively primet to 2, 3, and 5. In this case you get 29 and 31. Both happen to be prime numbers. We've found a twin.
The hard part of proving there are an infinite number of twins is finding a way of showing that your relatively prime numbers are truely prime. IE, in this case that neither is divisible by a higher prime such as 7, or 11.
Short answer: Yes, -3 is prime in the integers.
Long answer:
So 3|6 because 6=3*2, and (-3)|6 because 6 = (-3)*(-2), and so on.
As an example of this, 48 = 6*8. 3|48, so for 3 to be prime, we would need either 3|6 or 3|8.
Since 3|6, it passes the test in this case (but remember, the condition has to be true for all pairs of integers.)
since 3|a if and only if (-3)|a (Proof: Slashdot post is too small to contain it
Well... if your two primes are p-1 and p+1 you could use them as the primes in the RSA algorithm. I mean, it's not like it's trivial to break a composite number of the form p^2-1. :-)
(For those who don't know applied number theory, when factoring a number you first try the small primes, then assuming two large factors of the form (p-n)(p+n) = p^2-n^2.)
For every complex problem there is an answer that is clear, simple, and wrong. -- H L Mencken
From the Mercury News:
``Mathematicians described the advance -- announced at a conference in Germany -- as the most important breakthrough in the field in decades."
Oh, & proving Fermat's Last Theorem in 1995 was just another undergraduate exercise?
(For the non-math-nerds, proving true Fermat's Theorem -- that the formula a^n * b^n = c^n was insolveable where n is greater than 2 -- was considered for three _centuries_ to be _the_ principal challenge in mathematics. The man who did this -- Andrew Wiles -- spent about 30 years working on this, & succeeded only after a second try.)
Geoff
I think I see a trend here. Maybe for them it really would be easier to muzzle the entire internet than to produce p
Small gaps between consecutive primes
:= lim_inf(n -> +inf, (p_(n+1) - p_n) / log(p_n))) = 1
Recent work of D. Goldston and C. Yildirim
What are the shortest intervals between consecutive prime numbers? The twin prime conjecture, which asserts that p_(n+1) - p_n = 2 infinitely often is one of the oldest problems; it is difficult to trace its origins.
In the 1960's and 1970's sieve methods developed to the point where the great Chinese mathematician Chen was able to prove that for infinitely many primes p the number p + 2 is either prime or a product of two primes. However the well-known ``parity problem'' in sieve theory prevents further progress.
What can actually be proven about small gaps between consecutive primes? A restatement of the prime number theorem is that the average size of p_(n+1) - p_n is log(p_n) where p_n denotes the nth prime. A consequence is that
delta
In 1926, Hardy and Littlewood, using their ``circle method'' proved that the Generalized Riemann Hypothesis (that neither the Riemann zeta-function nor any Dirichlet L-function has a zero with real part larger than 1/2) implies that delta = 2/3. Rankin improved this (still assuming GRH) to delta = 3/5. In 1940 Erdös, using sieve methods, gave the first unconditional proof that delta 1. In 1966 Bombieri and Davenport, using the newly developed theory of the large sieve (in the form of the Bombieri - Vinogradov theorem) in conjunction with the Hardy - Littlewood approach, proved delta = 1/2 unconditionally, and then using the Erdös method they obtained delta = (2 + sqrt(3))/ 8 = 0.46650... . In 1977, Huxley combined the Erdös method and the Hardy - Littlewood, Bombieri - Davenport method to obtain delta = 0.44254. Then, in 1986, Maier used his discovery that certain intervals contain a factor exp(gamma) of more primes than average intervals. Working in these intervals and combining all of the above methods, he proved that delta = 0.2486, which was the best result until now.
Dan Goldston and Cem Yildirim have a manuscript which advances the theory of small gaps between primes by a quantum leap. First of all, they show that delta = 0. Moreover, they can prove that for infinitely many the inequality
p_(n+1) - p_n (log(p_n))^(8/9)
holds.
Goldston's and Yildirim's approach begins with the methods of Hardy-Littlewood and Bombieri - Davenport. They have discovered an extraordinary way to approximate, on average, sums over prime k-tuples. We believe, after work of Gallagher using the Hardy-Littlewood conjectures for the distribution of prime k-tuples, that the prime numbers in a short interval [N, N + lambda log(N)] are distributed like a Poisson random variable with parameter lambda. Goldston and Yildirim exploit this model in choosing approximations. They ultimately use the theory of orthogonal polynomials to express the optimal approximation in terms of the classical Laguerre polynomials. Hardy and Littlewood could have proven this theorem under the assumption of the Generalized Riemann Hypothesis; the Bombieri - Vinogradov theorem allows for the unconditional treatment.
This new approach opens the door for much further work. It is clear from the manuscript that the savings of an exponent of 1/9 in the power of log(p_n) is not the best that the method will allow. There are (at least) two possible refinements. One is in the examination of lower order terms that arise in his method. Can they be used to enhance the argument? The other is in the error term Gallagher found in summing the ``singular series'' arising from the Hardy-Littlewood k-tuple conjecture. There is reason to believe that this error term can be improved, possibly using ideas in recent work of Montgomery and Soundararajan (``Beyond Pair - Correlation''.)
It is not clear just how far this method can be pushed and what other problems might be attacked using his new ideas; at this point we can't rule ou
There is a small missunderstanding here. The smallest gap between two primes is 1 (between 2 and 3). But there is only this one pair of primes (2, 3) with such a small gap. But the next smallest gap 2 happens several times, between 3 and 5, 5 and 7, 11 and 13, 17 and 19 etc.pp.
So the question is: Does this happen at very large primes too? Are there primes between 10^50 and 10^55 which are just 2 apart? Note that the primes get more and more seldom, if you move at higher numbers.
The mathematic society knows since a long time that the distance between two neighbouring primes p(n) and p(n+1) is smaller than log(p(n)). But log(p(n)) grows also steadily, even much more slowly than p(n).
The mathematic society knows also that surprisingly small gaps between prime numbers happen von time to time. So the best number known until now was that there is an infinite number of primes p(n), where p(n+1) - p(n) was smaller than log(p(n)) * 0,22... (which grows also, but very, very slowly...)
The breaking news is, that now the factor 0,22... got improved to 0, which doesn't grow at all.
That means: there is an infinite number of primes, for which (p(n+1) - p(n))/log(p(n)) is quite close to 0.
This means: There are infinite often pairs of primes whose difference is quite small... It could easily be that the difference is 2 for an infinite number of pairs, which is not proved yet, but at least the minimal distance of two neighbouring primes does not grow beyond a very low number.