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

*To*: mathgroup at smc.vnet.net*Subject*: [mg52114] equal distribution of last digits base ten in the primes by b-normality*From*: Roger Bagula <tftn at earthlink.net>*Date*: Thu, 11 Nov 2004 04:52:21 -0500 (EST)*Reply-to*: tftn at earthlink.net*Sender*: owner-wri-mathgroup at wolfram.com

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

**Follow-Ups**:**Re: equal distribution of last digits base ten in the primes by b-normality***From:*Daniel Lichtblau <danl@wolfram.com>