# W-trick

(Difference between revisions)
 Revision as of 05:30, 5 July 2013 (view source)← Older edit Current revision (10:05, 5 July 2013) (view source)m (Reverted edits by Cheaptara (Talk) to last version by Teorth) Line 1: Line 1: - Be certain which you get only these screen sets that may viewable from any angle. + The W-trick is a simple trick to remove some of the structure from the set of primes, and also to increase the density of primes slightly. Basically, instead of looking at the set ${\mathcal P} := \{2,3,5,7,\ldots\}$ of primes, one instead looks at the set ${\mathcal P}_{W,b} := \{ n: Wn+b \in {\mathcal P} \}$, where $W := \prod_{p \leq w} p$ is the product of all primes less than some threshold w, and b is a number coprime to W (in many cases one can just take b=1).  It is a result in elementary number theory that $W = \exp((1+o(1))w)$, thus W is of exponential size in w.  If one seeks primes less than N, one must therefore set w no larger than $\log N$ or so. - [http://www.dvdmoviessaleuk.co.uk/ Buy dvd] , Find more information about Mac DVD to Audio Converter suggestions from Top Mac Application Online Shop.which includes simpsons,south park,Walt Disney's 100 Years Of Magic,Family Guy Seasons and so on,A portion of the goods is no cost shipping, purchase inexpensive dvds, A variety of of discounts attainable at dvdexcellent,welcome to our on line store. + The set ${\mathcal P}_{W,b} is a little bit denser than the primes; the primes less than N have density about [itex]1/\log N by the prime number theorem, but [itex]{\mathcal P}_{W,b}$ has density about $\frac{W}{\phi(W)} \frac{1}{\log N} \approx \frac{\log w}{\log N}$; raising w to $O(\log N)$ (which is the largest feasible value), the density of the W-tricked primes thus increases slightly from $1/\log N$ to about $\log \log N / \log N$. - [http://www.shopdvdmoviesonline.com/ Buy dvd] ,Practically nothing beats a rest day with household, mates or acquaintances. + The W-tricked primes [itex]{\mathcal P}_{W,b} also behave more 'pseudorandomly' than the primes themselves.  For instance, the primes are of course highly biased to favour odd numbers over even numbers, but the W-tricked primes have no such bias once w is at least 2 (by the prime number theorem in arithmetic progressions).  More generally, the primes do not equidistribute modulo q for any q, but the W-tricked primes do as soon as w is greater than or equal to all the prime factors of q. - + - [http://www.populardvdonline.com/ Dvd] , Although headrest car DVD player isn't the coolest and most entertaining way for you to pimp your car or truck but to make your time within the road much more enjoyable. + - + - [http://www.dvdshopaustralia.com/ Buy dvd] , Incorporated in the package are AC/DC adaptor, AV cable, auto charger and headrests mounting Velcro strap. Installing this necessary leisure device will make your travel relaxing. Shops frequently give a restricted alternative as well as expense are often a little bit increased when compared with what companies give. Among the reasons for that is definitely that despite the fact that they change their all round design consistently, the color, background as well as the dimensions of the symbol always remain exactly the same. For years, he had brought the benefits of the most recent technologies inside the DVD marketplace. + - + - [http://www.dvdfortvshowes.com/ Blu ray dvds] , Getting this kind of accessory is really a assure for any additional comfortable trip. We promise that all the postage might be charged by your actual product's weight, there won't be any future extra charge: insurance&tax. +

## Current revision

The W-trick is a simple trick to remove some of the structure from the set of primes, and also to increase the density of primes slightly. Basically, instead of looking at the set ${\mathcal P} := \{2,3,5,7,\ldots\}$ of primes, one instead looks at the set ${\mathcal P}_{W,b} := \{ n: Wn+b \in {\mathcal P} \}$, where $W := \prod_{p \leq w} p$ is the product of all primes less than some threshold w, and b is a number coprime to W (in many cases one can just take b=1). It is a result in elementary number theory that W = exp((1 + o(1))w), thus W is of exponential size in w. If one seeks primes less than N, one must therefore set w no larger than logN or so.

The set ${\mathcal P}_{W,b}$ is a little bit denser than the primes; the primes less than N have density about 1 / logN by the prime number theorem, but ${\mathcal P}_{W,b}$ has density about $\frac{W}{\phi(W)} \frac{1}{\log N} \approx \frac{\log w}{\log N}$; raising w to O(logN) (which is the largest feasible value), the density of the W-tricked primes thus increases slightly from 1 / logN to about loglogN / logN.

The W-tricked primes ${\mathcal P}_{W,b}$ also behave more 'pseudorandomly' than the primes themselves. For instance, the primes are of course highly biased to favour odd numbers over even numbers, but the W-tricked primes have no such bias once w is at least 2 (by the prime number theorem in arithmetic progressions). More generally, the primes do not equidistribute modulo q for any q, but the W-tricked primes do as soon as w is greater than or equal to all the prime factors of q.