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MathGroup Archive 2004

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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: [mg52173] Re: equal distribution of last digits base ten in the primes by b-normality
  • From: Roger Bagula <tftn at earthlink.net>
  • Date: Sun, 14 Nov 2004 04:30:09 -0500 (EST)
  • References: <cmve1q$sgk$1@smc.vnet.net>
  • Reply-to: tftn at earthlink.net
  • Sender: owner-wri-mathgroup at wolfram.com

-------- Original Message --------
Subject: [mg52173] Re: [numbertheory] equal distribution of last digits base ten 
in the primes b...
From: mikeoakes2 at aol.com
To: mathgroup at smc.vnet.net
Reply-To: numbertheory at yahoogroups.com



See my post
http://listserv.nodak.edu/scripts/wa.exe?A2=ind0403&L=nmbrthry&P=R1117&D=0&H=0&O=T&T=1 
<http://listserv.nodak.edu/scripts/wa.exe?A2=ind0403&L=nmbrthry&P=R1117&D=0&H=0&O=T&T=1>
 
for counts of primes < 10^13:-
 
[8,1]=86516283187
[8,3]=86516411563
[8,5]=86516425996
[8,7]=86516416092

-Mike Oakes

 
In a message dated 10/11/2004 23:07:38 GMT Standard Time, 
tftn at earthlink.net writes:

    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 

 



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