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• ### Re: [PrimeNumbers] Helfgott's improvement of Erathostenes' sieve

...37 GMT+02:00 Alan Eliasen eliasen@^\$1 [primenumbers] < primenumbers-noreply@^\$2 > : On 2016-10-10 01:39, Jose Ramón Brox López ambroxius@^\$3 [primenumbers] wrote: > There is news these days that Helfgott's has found an algorithm to reduce > the space complexity of...
Jose Ramón Brox López Oct 10, 2016
• ### Helfgott's improvement of Erathostenes' sieve

There is news these days that Helfgott's has found an algorithm to reduce the space complexity of Erathostenes' sieve (e.g. https://www.scientificamerican.com/article/new-take-on-an-ancient-method-improves-way-to-find-prime-numbers/ ). Anyone knows where to find some related paper, or has attended any conference and can explain the main idea here? Warmest regards! Jose Brox
Jose Ramón Brox López Oct 10, 2016
• ### Re: [PrimeNumbers] How could one show without computers that there was no number x^2-y^2 that equalled R19

Hi jp! I don't know if I am understanding you well. The only natural numbers that cannot be written as a difference of two squares are those of the form 4n+2, i.e., the numbers which are even but not divisible by 4; the rest of them are all differences of two squares. In particular, if n=2k+1 is an odd number, then (k+1)^2-k^2 = 2k+1 = n. Therefore R19 IS a difference of two...
Jose Ramón Brox May 29, 2014

Dear sinclair75: In ill-conditioned systems and chaotic systems , precision is paramount (because a little error in the input data produces a huge error in the output data or in the extrapolation to future steps, respectively). I think a good example is weather forecasting. Nevertheless, approaches to avoid this undesirable situation do exist: for example, an infinite-digits...
Jose Ramón Brox Mar 8, 2014
• ### Re: [PrimeNumbers] Re: Polynomials

2013/7/26 djbroadhurst > ** > > > > > --- In primenumbers@^\$1, > Jose Ram�n Brox wrote: > > > Now that I think about it better, > > I see that I was fooled by your proposal! > > Indeed you were :-) > > Yeah, you are so right! Hehe You can see that I don't know that much Pari-GP programming; I thought that "polinterpolate" of the vector V was interpolation at (V, [0 ... 0]), but...
Jose Ramón Brox Jul 27, 2013
• ### Re: [PrimeNumbers] Re: Polynomials

2013/7/27 Jose Ram�n Brox > > (But if you paste David's program on Pari and correct the missing }, you > will get a glimpse of it, if so you wish). > > What I really wanted to say is: if you paste David's _polynomial_ and add the _necessary_ }, you will get it. You must also erase the ";", like this: {Poly=polinterpolate(vector(100,k,prime(137*k-104)))} > Jose > > > >> Werner...
Jose Ramón Brox Jul 27, 2013
• ### Re: [PrimeNumbers] Re: Polynomials

2013/7/26 Alexander > ** > > > > > I don't have your math program. > That's Pari-GP, Alexander. You can get it here: http://pari.math.u-bordeaux.fr/ (It seems not to be working right now). Can I get your P(n) explicitly? > Do you really want us to paste here a polynomial of degree 99 with big rational coefficients? To see what I'm saying, its independent term is...
Jose Ramón Brox Jul 27, 2013
• ### Re: [PrimeNumbers] Re: Polynomials

Now that I think about it better, I see that I was fooled by your proposal! :D What we want is not a polynomial whose _roots_ are primes, what we want is a polynomial which generates as much primes as possible with its _image_. I.e., what we want is to maximize p such that the sequence p({1,2,3,...}) gives the biggest possible "prime head". The classical example of Euler, p(x)=x^2...
Jose Ramón Brox Jul 26, 2013
• ### Re: [PrimeNumbers] Re: Polynomials

2013/7/26 djbroadhurst > ** > This polynomial performs better: > > polinterpolate(vector(100,k,prime(k))) > > Sure! But it (I suppose) does not satisfy the implicit assumption (explicit in the contest) that the primes must be consecutive. Regards, Jose [Non-text portions of this message have been removed]
Jose Ramón Brox Jul 26, 2013