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

*To*: mathgroup at smc.vnet.net*Subject*: [mg52203] Re: equal distribution of last digits base ten in the primes by b-normality*From*: drbob at bigfoot.com (Bobby R. Treat)*Date*: Sun, 14 Nov 2004 04:31:24 -0500 (EST)*References*: <cmve1q$sgk$1@smc.vnet.net> <cn4luo$187$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

>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 Gauss knew the mean density of primes by 1792 and Riemann's hypothesis was stated in 1859, so the Kronecker theorem cited below can scarcely be said to predate that knowledge -- even if the PNT proof didn't come until 1896. Nor is it particularly surprising, anyway, that one might prove four densities equal without knowing what they are. Nor does this have anything to do with b-normality, which has to do with the distribution of blocks of digits in ONE irrational number's infinite expansion base b. The cited theorem involves the modulus of DIFFERENT prime numbers mod b. Besides, 10-normality requires all ten digits 0-9 to appear equally often in the decimal expansion of an irrational, not just 1, 3, 7, and 9. Bobby Roger Bagula <tftn at earthlink.net> wrote in message news:<cn4luo$187$1 at smc.vnet.net>... > Forward from another list : a little known proof does exist. > ( Thanks to Peter Pleasants library research we know about it now) > > -------- Original Message -------- > Subject: [mg52203] 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 > > -------------------------------------------------------------------- > mail2web - Check your email from the web at > http://mail2web.com/ . > > > > > -- > 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 > > > > > >