Re: [PrimeNumbers] Euclidean generation of prime numbers
- Thanks for all the replies.
Bhupinder Singh Anand (R) wrote:
>On Thursday, February 01, 2007 4:17 AM, Alastair Farrugia wrote in--
>AF>> ... Is there any literature about the properties of this algorithm
>Years ago, I investigated similar sieve algorithms, which you can find
>Three Theorems on modular sieves that suggest the Prime Difference is
>O(Number of primes < (p(n)^1/2))
One must live the way one thinks,
or end up thinking the way one has lived.
Paul Bourget (1852 - 1935)
Freedom is never voluntarily given by the oppressor;
it must be demanded by the oppressed.
Martin Luther King Jr.
- For example 2 adjacent gaps cannot be equal if they aren't multiple of
6. For example the gap between 2 pairs of twins is at least 4. For
example each prime number has the form 2n+/-1, 3n+/-1, 4n+/-1, 6n+/-1.
Each pair of twins has the form 12n+-1, there are approximate formulas
for the nth prime and the number of primes < x and so on. You cannot
call all this random ore unpredictable. Of course the prime numbers are
distributed as regularly as possible, that's a tautology. In
mathematics everything is as regular as possible. Is pi random? Build
P=2,357111317192329 , and you have the same case as pi. Consider the
primes to be an irrational number, and there are no problems. If you
mean there is no formula f(n) which produces primes for each n, then
you are right. In this sense primes are random. (I am not quite sure
there is a formula p=[k^n^3] (H.W.Mills) which is said to produce only
prime numbers). If you define "formula" as an algorithm, as a
calculation instruction such as the sieve of Eratosthenes, then the
primes are not random but simply what they are. Perhaps the compound
numbers are random? Or are they only non-transparently complicated?