John,

On 26 May 2013, at 00:54, John Mikes wrote:

Bruno and others:
did you read

http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.single.html

the information about prof. Zhang's discovery (U of New Hampshire)?
It is still in the conjecture of mathematical proof and 'truth' with a position of "primes are greater than 1" - with the interesting conclusion that 'primes' are the ATOMS of the number world.
Any thoughts?

Primes (1 is usually not considered as a prime number) are atoms of the numbers when conceived multiplicatively, because all numbers can be described uniquely as a product of primes. That is the existence and unicity of decomposition of numbers into prime factors (without taking the order of the multiplication into account). This is the so called fundamental theorem of arithmetic. It is easy to prove the existence of the decomposition into primes, but less easy to prove the uniqueness.

For the twin conjecture, (it exists an infinity of pair of primes p and q with p - q = 2) it looks like an important step has been proved, (the case with p - q just bounded) but we are still far from proving the twin one. Most mathematician believe that the twin conjecture is true (like most believe that the Riemann conjecture is true). If they were false, the distribution of primes would not be "statistically random", and that would mean something very special is at play, a bit like a number conspiracy! Why not, of course. We just don't know, but a non random behavior of the primes is a bit like the UFO of number theory. Well, except that for the UFO, there are (at least) some evidences (from time to time, most are eventually explained in general), but there is no evidence at all that the primes behave non- randomly (in the statistical sense, not in Chaitin-Kolmogorov sense as we can generate mechanically the distribution of primes).

Bruno



http://iridia.ulb.ac.be/~marchal/



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