Antigravics

http://www.geocities.com/~harveyh/primes.htm

Pairs equal Prime


Priming the Cube

http://www.ieeta.pt/~tos/gaps.html
Gaps between consecutive primes

http://primes.utm.edu/glossary/page.php ... Conjecture

Goldbach wrote a letter to Euler dated June 7, 1742 suggesting (roughly) that every even integer is the sum of two integers p and q where each of p and q are either one or odd primes. Now we often word this as follows:

Goldbach's conjecture: Every even integer n greater than two is the sum of two primes.

This is easily seen to be equivalent to

Every integer n greater than five is the sum of three primes.

There is little doubt that this result is true, as Euler replied to Goldbach:

That every even number is a sum of two primes, I consider an entirely certain theorem in spite of that I am not able to demonstrate it.

Progress has been made on this problem, but slowly--it may be quite awhile before the work is complete. For example, it has been proven that every even integer is the sum of at most six primes (Goldbach suggests two) and in 1966 Chen proved every sufficiently large even integer is the sum of a prime plus a number with no more than two prime factors (a P2).

Vinogradov in 1937 showed that every sufficiently large odd integer can be written as the sum of at most three primes, and so every sufficiently large integer is the sum of at most four primes. One result of Vinogradov's work is that we know Goldbach's theorem holds for almost all even integers.

Various heuristic estimates are avaliable for the number of solutions there should be to 2n = p + q (with p, q prime). And these of course grew in size with n.

Recently, Jean-Marc Deshouillers, Yannick Saouter, and Herman te Riele have verified this up to 1014 with the help of a Cray C90 and various workstations. If we will accept the Riemann Hypothesis, then this is enough to prove the odd Goldbach conjecture: Every odd integer greater than five is the sum of three primes.

Descartes also was aware of the two prime version of Goldbach's conjecture before Goldbach was. So is it misnamed? Erdös said that it "is better that the conjecture be named after Goldbach because, mathematically speaking, Descartes was infinitely rich and Goldbach was very poor."

When verifying the Goldbach conjecture for n we quickly see that it is very easy to find many primes which add to n. In 1923 Hardy and Littlewood took the first major step toward the proof of the Goldbach conjectures using their circle method [HL23]. Among other things, they conjectured that the number of ways of writing n as the sum of two primes, G(n), is asymptotic to twice the twin prime constant times n/(log n)2 times the product of (p-1)/(p-2) taken over the prime divisors p of n.

That is a lot of solutions for a large n so there will be a solution with one of the prime quite small. For example, when checking all n up to 100,000,000,000,000, Richstein [Richstein2000] found the smaller prime is never larger than 5569 used in the following partition:

389965026819938 = 5569 + 389965026814369.

It was conjectured [GRL89] that the smallest prime in a partition of n might be bounded by a constant times (log n)2 log log n. Richstein's work shows that the constant 1.603 (from n=6) sufficies for n < 4.1014.

2n = p + q for all n > 1. p,q prime.

http://www.physorg.com/news114363930.html

>> French "mathlete" Alexis Lemaire showed off his rare mental agility Thursday, claiming a new world record after working out in his head the 13th root of a random 200-digit number in just 72.4 seconds.

Lemaire, a 27-year-old doctoral student in artificial intelligence from Reims, near Paris, sat at a laptop computer that randomly selected the figure and displayed it on the screen. The number was so long it ran over 17 lines.

"The first digit is very easy, the last digit is very easy, but the inside numbers are extremely difficult," the mental gymnast told AFP after the performance at New York's Hall of Science.

"I use an artificial intelligence system which I use on my own brain instead of on a computer," he explained, matter-of-factly.

"Personally, I believe most people can do it but I have also a high-speed mind. My brain works sometimes very, very fast."

Pressing his point, he adds: "Sometimes when I do multiplication my brain works so fast that I need to take medication."

"I think somebody without a very fast brain can also do this kind of multiplication but this is maybe easier for me because my brain is faster."

Lemaire says he first realized he had a knack for numbers when he was around 11 years old, but perhaps surprisingly he did not do well in maths at school.

"I was not top of the class. I was an autodidact, mostly by books," he says.

"I use a process to improve my skills, to behave like a computer. When I do something wrong, I learn from that," he says. "It's like running a program in my head ... to control my brain," he says. >>