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Re: equal distribution of last digits base ten in the primes by b-normality
*To*: mathgroup at smc.vnet.net
*Subject*: [mg52142] Re: [mg52114] equal distribution of last digits base ten in the primes by b-normality
*From*: Daniel Lichtblau <danl at wolfram.com>
*Date*: Fri, 12 Nov 2004 02:13:56 -0500 (EST)
*References*: <200411110952.EAA28808@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
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
>
I fail to see this.
"On the random character of fundamental constant expressions" (2000) by
David H. Bailey and Richard E. Crandall shows that (among many other
things), subject to a certain hypothesis about a class of iterated map,
Pi is normal to base 16. Perhaps further work has been done in this area
since then. My very limited understanding from that article is that they
proved that for Pi the map does not have a finite attractor and hence
the second case of the hypothesis can be used.
I do not see how the iterations above fall into that hypothesis, or how
one might prove there is no finite attractor.
Daniel Lichtblau
Wolfram Research
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