Generate a list in Mathematica with a conditional tested for each element

Question

Suppose we want to generate a list of primes p for which p + 2 is also prime.

A quick solution is to generate a complete list of the first n

Answer 1

Here's another couple of alternatives using NextPrime:

pairs1[pmax_] := Select[Range[pmax], PrimeQ[#] && NextPrime[#] == 2 + # &]

pairs2[pnum_] := Module[{p}, NestList[(p = NextPrime[#];
                      While[p + 2 != (p = NextPrime[p])]; 
                      p - 2) &, 3, pnum]] 

and a modification of your Reap/Sow solution that lets you specify the maximum prime:

pairs3[pmax_] := Module[{k,p},
                   Reap[For[k = 1, (p = Prime[k]) <= pmax, k++,
                        If[PrimeQ[p + 2], Sow[p]]]][[-1, 1]]]

The above are in order of increasing speed.

In[4]:= pairs2[10000]//Last//Timing
Out[4]= {3.48,1261079}
In[5]:= pairs1[1261079]//Last//Timing
Out[5]= {6.84,1261079}
In[6]:= pairs3[1261079]//Last//Timing
Out[7]= {0.58,1261079}