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>
• 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
>
>alternative email: rlbtftn at netscape.net
>
>
>

--
Respectfully, Roger L. Bagula