W-trick

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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.

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