1,1

The largest presently known value of prime(10^n) is

prime(10^18) = 44211790234832169331 this compares to

primex(10^18) = 44211790234127235727 accurate to 11 places

Here the sign of prime(x)-primex(x) is positive. However the sign changes as x varies. The following is a table with the relative error and sign change:

n, prime(10^n), primex(10^n), relative error

6, 15485863, 15484040, 1.1772 E-4

7, 179424673, 179431239, -3.6594 E-4

8, 2038074743, 2038076587, -9.0478 E-5

9, 22801763489, 22801797576, -1.4949 E-5

10, 252097800623, 252097715777, 3.3655 E-6

11, 2760727302517, 2760727752353, -1.6294 E-6

12, 29996224275833, 29996225393465, -3.7259 E-7

13, 323780508946331, 323780512411510, -1.0702 E-7

14, 3475385758524527, 3475385760290723, -5.0820 E-8

15, 37124508045065437, 37124508056355511, -3.0411 E-9

16, 394906913903735329, 394906913798224969, 2.6718 E-9

17, 4185296581467695669, 4185296581676470048, -4.9883 E-11

18, 44211790234832169331,44211790234127235727, 1.5944 E-11

Primex(n) ~ prime(n). Prime(n) is the n-th prime number. Primex(n) is the Riemann-Gram approximation of Prime(n) accurate to log10(n)/2 + 1 digits for large n. The sequence is primex(A007097(n)) for n = 1 to 18.

A007097(17) = 75063692618249

Primex(75063692618249) = 2586559741900744

A007097(18) = 2586559730396077

Primex(2586559730396077) = 98552043877145945

A007097(19) ~ 98552043800000000

(PARI) xeqprimex(n) = { local(a, x); a = [1, 2, 3, 5, 11, 31, 127, 709, 5381, 52711, 648391, 9737333, 174440041, 3657500101, 88362852307, 2428095424619, 75063692618249, 2586559730396077]; for(x=1, n, print1(round(primex(a[x]))", ") ) } \Approximates the n-th prime number to an accuracy of log10(n)/2 places. primex(n) = { local(x, px, r1, r2, r, p10, b, e, est); if(n==1, return(2)); \ force to 2 b=10; \Select base p10=log(n)/log(10); \Determine p10 = power of 10 of n to adjust b^p10 if(Rg(b^p10*log(b^(p10+1)))< b^p10, m=p10+1, m=p10); r1 = 0; r2 = 7.718281828; \Real kicker. if r2=1, it bonks at 1e117 for(x=1, 100, r=(r1+r2)/2; est = (b^p10*log(b^(m+r))); px = Rg(est); if(px <= b^p10, r1=r, r2=r); r=(r1+r2)/2; ); est; } Rg(x) = \Gram's Riemann Approx of Pi(x) { local(n=1, L, s=1, r); L=r=log(x); while(s<10^40*r, s=s+r/zeta(n+1)/n; n=n+1; r=r*L/n); (s) }

Cf. A007097.

Sequence in context: A007097 A132745 A124538 this_sequence A064095 A061935 A067078

Adjacent sequences: A124624 A124625 A124626 this_sequence A124628 A124629 A124630

nonn,uned

Cino Hilliard (hillcino368(AT)hotmail.com), Dec 21 2006

a(19) and a(20) from David Baugh (dbaugh(AT)rice.edu), Oct 25 2007