Hello,  there is a little jewel of program called primecount (easy to find on Windows, mac or linux). It counts the primes. Here is a table of results (see below), from 10^1 to 10^20 with primecount and mathematica : mathematica stops giving answers after 10^14 and takes 26.2656 seconds to find PrimePi[10^14] when primecount takes only : 0.069 seconds (!). This is 431 times faster. I do not mention Maple for decency, it is too slow by far, Pari-Gp is not bad but limited in range too. The -g option stands for the Gourdon method (apparently the fastest). You can download the program here : https://github.com/kimwalisch/primecount it comes with a short but very useful explanation. See for yourself : results are astonishing. The recent record result of Pi(10^27) is mainly due to this program. See http://oeis.org/A006880 for more details. Last thing : the program is 1.52 megs in size and takes all the cpus/threads you may have on one machine automatically. primecount -g 10^1 --time; 4 seconds: 0.000 primecount -g 10^2 --time; 25 seconds: 0.000 primecount -g 10^3 --time; 168 seconds: 0.000 primecount -g 10^4 --time; 1229 seconds: 0.000 primecount -g 10^5 --time; 9592 seconds: 0.000 primecount -g 10^6 --time; 78498 seconds: 0.000 primecount -g 10^7 --time; 664579 seconds: 0.000 primecount -g 10^8 --time; 5761455 seconds: 0.001 primecount -g 10^9 --time; 50847534 seconds: 0.001 primecount -g 10^10 --time; 455052511 seconds: 0.003 primecount -g 10^11 --time; 4118054813 seconds: 0.007 primecount -g 10^12 --time; 37607912018 seconds: 0.015 primecount -g 10^13 --time; 346065536839 seconds: 0.026 primecount -g 10^14 --time; 3204941750802 seconds: 0.069 primecount -g 10^15 --time; 29844570422669 seconds: 0.204 primecount -g 10^16 --time; 279238341033925 seconds: 0.647 primecount -g 10^17 --time; 2623557157654233 seconds: 2.276 primecount -g 10^18 --time; 24739954287740860 seconds: 8.334 primecount -g 10^19 --time; 234057667276344607 seconds: 35.579 primecount -g 10^20 --time; 2220819602560918840 seconds: 155.561 Timing[PrimePi[10^1]] {0., 4} Timing[PrimePi[10^2]] {0., 25} Timing[PrimePi[10^3]] {0., 168} Timing[PrimePi[10^4]] {0., 1229} Timing[PrimePi[10^5]] {0., 9592} Timing[PrimePi[10^6]] {0., 78498} Timing[PrimePi[10^7]] {0., 664579} Timing[PrimePi[10^8]] {0., 5761455} Timing[PrimePi[10^9]] {0., 50847534} Timing[PrimePi[10^10]] {0.046875, 455052511} Timing[PrimePi[10^11]] {0.1875, 4118054813} Timing[PrimePi[10^12]] {1.03125, 37607912018} Timing[PrimePi[10^13]] {5.34375, 346065536839} Timing[PrimePi[10^14]] {26.2656, 3204941750802} Have fun !,  Simon Plouffe