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

• To: mathgroup at smc.vnet.net
• Subject: [mg52114] equal distribution of last digits base ten in the primes by b-normality
• From: Roger Bagula <tftn at earthlink.net>
• Date: Thu, 11 Nov 2004 04:52:21 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```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