Re: equal distribution of last digits base ten in the primes by b-normality

*To*: mathgroup at smc.vnet.net*Subject*: [mg52164] Re: equal distribution of last digits base ten in the primes by b-normality*From*: Roger Bagula <tftn at earthlink.net>*Date*: Sat, 13 Nov 2004 04:40:13 -0500 (EST)*References*: <cmve1q$sgk$1@smc.vnet.net>*Reply-to*: tftn at earthlink.net*Sender*: owner-wri-mathgroup at wolfram.com

(* Modulo 10 prime gaps as a product function to get primes*) w[n_]=Abs[Mod[Prime[n+1],10]-Mod[Prime[n],10]] p[n_]=1+Product[w[m],{m,2,n}] digits=200 a=Delete[Union[Table[If[PrimeQ[p[n]]==True,p[n],0],{n,1,digits}]],1] Roger Bagula wrote: >The {1,3,7,9} last digits of the primes modulo 10 >equal distribution >conjecture has never been proved, >but I have a b- normal iteration for it.. >What that says is that the modulo ten function >is equally spaced over the base ten. >This is the same argument that Dr. Bailey used to >say that the digits of Pi are equally probable over base 16 >using his Pi digits formula. >Thus if Bailey's proof is acceptable so is this. >So with experimental evidence of several million primes >and this type of functional evidence/proof >it has been pretty well estsablished that the four last digits appear >equally. > >Clear[x,a,digits,f] >(* designed covergent sum and b- normal iterator based on the Prime >first digits modulo 10*) >(* sorted iterative randoms form a devil's staircase like step *) >f[n_]=1/((10-Mod[Prime[n],10])*10^n) >digits=200 >a=Table[N[f[n],digits],{n,1,digits}]; >b=N[Apply[Plus,a],digits] >x[n_]:=x[n]=Mod[10*x[n-1]+1/(10-Mod[Prime[n],10]),1] > x[0]=0 >Clear[a,b] >a=Table[N[x[n],digits],{n,0,digits}]; >ListPlot[a,PlotJoined->True,PlotRange->All] >b=Sort[Table[N[x[n],digits],{n,0,digits}]]; >ListPlot[b,PlotJoined->True,PlotRange->All] >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 > > > -- 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