Re: 100 digit base ten primes

*To*: mathgroup at smc.vnet.net*Subject*: [mg52259] Re: 100 digit base ten primes*From*: drbob at bigfoot.com (Bobby R. Treat)*Date*: Thu, 18 Nov 2004 01:44:59 -0500 (EST)*References*: <cneusb$qc2$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

The code below demonstrates several things: 1) Yes, there's at least one prime between n and n + Log[n]^2. In fact, there are MANY primes in the range when n is large. 2) The average of those limits (rounded to an integer) almost never delivers a prime number. (It didn't in your own examples.) Exactly half the time it's even, for instance. 3) Searching the entire range is silly if you're searching for a prime; why not stop at the first one you find? 4) The upper bound (and the theorem itself) is irrelevant to a practical search for primes. It puts a bound on how far you'll have to search, but knowing that doesn't change the search method in any way. On another note, Dimensions[a][[1]] is usually computed as Length@a, and evaluating PrimeQ[m+Floor[Log[m]^2/2]]==True is the same result as evaluating PrimeQ[m+Floor[Log[m]^2/2]] That's False in both cases from your post. Here's a test for exponents ranging from 1 to 100: f[n_] = n + Floor[Log[n]^2/2]; g[exp_] = 2^Prime[exp] - 1; select[exp_] := If[PrimeQ[f[g[exp]]], f[g[exp]], Sequence @@ {}] select /@ Range[100] Length[%] {3, 8231, 2361183241434822608057, 3064991081731777716716694054300618367237478244367212221} 4 That's four prime numbers in a hundred tries, no better than random, so let's try replacing f with something else: f[n_] = 2*n - 1; select /@ Range[100] Length[%] {5, 13, 61, 4093, 16381, 1048573, 16777213, 14272476927\ 05959881058285969449495136382746\ 621, 239452428260295134118491722\ 99223580994042798784118781} 9 Or better yet: f[n_]=Prime@n; select/@Range@13 Length@% {5,17,127,709,17851,84011,1742527,7754017,148948133,11891268397, 50685770143,3839726846299,67756520645293} 13 Thirteen out of thirteen. The fourteenth candidate is too large for Mathematica's implementation of Prime. Bobby Roger Bagula <tftn at earthlink.net> wrote in message news:<cneusb$qc2$1 at smc.vnet.net>... > Clear[a,b,m,m0,m1,m3] > (* program for finding primes near 100 digits long base 10 using Mersenne > seed points*) > (* cryptography length primes*) > (* the logically most probable way someone taught*) > (* traditional number theory might use find 100 digits primes*) > (* short of using a sieve to 100 decimal places*) > Table[N[Log[2^Prime[n]]/Log[10]],{n,1,68}] > $MaxPrecision=Floor[Log[2^Prime[68]]/Log[10]]+1 > m=2^Prime[67]-1 > m0=Floor[m+Log[m]^2] > (m0-m)/2 > m1=2^Prime[68]-1 > m3=Floor[m1+Log[m1]^2] > (m3-m1)/2 > a=Delete[Union[Table[If[PrimeQ[n]==True,n,0],{n,m,m0,2}]],1] > Dimensions[a][[1]] > (* 1/Log[n] probability test*) > N[Dimensions[a][[1]]/((m0-m)/2)-1/Log[m0]] > b=Delete[Union[Table[If[PrimeQ[n]==True,n,0],{n,m1,m3,2}]],1] > Dimensions[b][[1]] > N[Dimensions[b][[1]]/((m3-m1)/2)-1/Log[m3]] > PrimeQ[m+Floor[Log[m]^2/2]]==True > PrimeQ[m1+Floor[Log[m1]^2/2]]==True > > Respectfully, Roger L. Bagula > > tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : > alternative email: rlbtftn at netscape.net > URL : http://home.earthlink.net/~tftn