Goldston's work is far more profound and interesting than most of the commenters seem to
realize, which is to be expected, given the level of ignorance of the average poster.
First of all, Goldston (and that Turkish chap) did not ``discover''
any new pattern among the primes -- mathematicans have long conjected that the prime numbers get very close together infinitely often. Indeed, the
so called Twin Prime Conjecture asserts that there are infinitely many consecutive primes p,q
(p p), such that
q - p k*ln(q) ? What is this question getting
at? Well, according to the Prime Number Theorem,
which was proved in the late 19th century, the number of primes in [2,x] is, asymptotically,
x/ln(x). Another way of saying the same thing is
that the average gap between consecutive primes in
[2,x] is about ln(x). Well, so this begs the question: How often are the gaps between consecutive primes in [2,x] much smaller than
this average (that is, ln(x)), and how often are they much larger than the average? This is
the source of the ``q-p k*ln(q)'' question above.
A whole string of mathematicians worked on this
small gaps question for decades, and the best result prior to Goldston's, which was due to H. Maier, was that there are infinitely many
consecutive primes p,q such that q - p 0.2 ln(q)
(actually, the constant here is a little more than 0.2). Using an ingenious new idea (I have read Goldston's paper) that combines
approximation theory, a method called amplification, the Bombieri-Vinogradov theorem,
as well as approximate von Mangoldt functions, Goldston has proved this short gaps conjecture (that is, one can replace the 0.2 with any number less than 1). In fact,
he proved a considerably stronger statement:
He showed that infinitely often
q-p (ln q)^{8/9}.
It seems clear that Goldston's work will have a
profound and long-lasting impact on prime number theory, and some people believe that it can be used to prove a $3,000 conjecture of P. Erdos
(although Erdos died a few years ago, you can still get money if you solve one of his prize problems) on large gaps between primes. Perhaps
it will one day lead to the solution of the
Twin Prime Conjecutre, or maybe even the Goldbach conjecture.
Please do not email me.
Slashdot Mirror
Goldston's work is far more profound and interesting than most of the commenters seem to realize, which is to be expected, given the level of ignorance of the average poster. First of all, Goldston (and that Turkish chap) did not ``discover'' any new pattern among the primes -- mathematicans have long conjected that the prime numbers get very close together infinitely often. Indeed, the so called Twin Prime Conjecture asserts that there are infinitely many consecutive primes p,q (p p), such that q - p k*ln(q) ? What is this question getting at? Well, according to the Prime Number Theorem, which was proved in the late 19th century, the number of primes in [2,x] is, asymptotically, x/ln(x). Another way of saying the same thing is that the average gap between consecutive primes in [2,x] is about ln(x). Well, so this begs the question: How often are the gaps between consecutive primes in [2,x] much smaller than this average (that is, ln(x)), and how often are they much larger than the average? This is the source of the ``q-p k*ln(q)'' question above. A whole string of mathematicians worked on this small gaps question for decades, and the best result prior to Goldston's, which was due to H. Maier, was that there are infinitely many consecutive primes p,q such that q - p 0.2 ln(q) (actually, the constant here is a little more than 0.2). Using an ingenious new idea (I have read Goldston's paper) that combines approximation theory, a method called amplification, the Bombieri-Vinogradov theorem, as well as approximate von Mangoldt functions, Goldston has proved this short gaps conjecture (that is, one can replace the 0.2 with any number less than 1). In fact, he proved a considerably stronger statement: He showed that infinitely often q-p (ln q)^{8/9}. It seems clear that Goldston's work will have a profound and long-lasting impact on prime number theory, and some people believe that it can be used to prove a $3,000 conjecture of P. Erdos (although Erdos died a few years ago, you can still get money if you solve one of his prize problems) on large gaps between primes. Perhaps it will one day lead to the solution of the Twin Prime Conjecutre, or maybe even the Goldbach conjecture. Please do not email me.