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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*: [mg52169] 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 04:40:20 -0500 (EST)
*References*: <cmve1q$sgk$1@smc.vnet.net>
*Reply-to*: tftn at earthlink.net
*Sender*: owner-wri-mathgroup at wolfram.com
Forward from another list : a little known proof does exist.
( Thanks to Peter Pleasants library research we know about it now)
-------- Original Message --------
Subject: [mg52169] Re: Last digits of primes
From: "peterpleasants at iprimus.com.au" <peterpleasants at iprimus.com.au>
To: mathgroup at smc.vnet.net
Reply-To: The Tiling Listserv <listserv at tiling.uttyler.edu>
{ other stuff deleted}
As promised, I went to look at L.E. Dickson's "History of the theory of
numbers" Vol. 1 (first published in 1919 by the Carnegie Institute of
Washington and reprinted in 1999 by AMS Chelsea Publishing). He cites
L. Kronecker as having proved in "Vorlesungen uber Zahlentheorie, I"
(1901) that "there is the same mean density of primes in each of the
phi(m) progressions mh+r_i, where the r_i are integers < m and prime
to m". What really surprised me about this is that this book is the
notes of lectures given in 1875-6, long before the proof of the prime
number theorem. That implies that the densities were known to be
equal even before it was known precisely what they were. Dickson also
ascribes a proof to Ch. de la Vallee-Poussin in Annales de la soc. sci.
de Bruxelles, 20, 1896, II, pp 281-361. That's more what I would
have expected: de la Vallee-Poussin was one of the two people who
independently proved the prime number theorem in 1896 and his proof of
PNT is on pp 183-256 of the same issue. For good measure, among several
other proofs Dickson cites, there is one by E. Landau in 1908.
The case that's been discussed on this list, of course, is m = 10 with
the r_i's 1,3,7,9.
Peter Pleasants
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--
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
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
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