Abstract: Experiments using Batrachian sequences suggested a fractal nature to the prime sequence that had a breaking wave effect. In this short article a method using a Mandelbrot cartoon in a Bescovitch-Ursell function in the sequential equation is presented with a low ramp three line cartoon. 

Body: This article is presented only as a method and not as a exact simulation, but the indications are that with the proper parameterizations and dimension a prime predictor of great accuracy can be achieved. A Batrachian behaves much like a Mandelbrot cartoon multifractal. I had done work in which I had gotten a Sierpinski gasket set using such multifractals in the past. Mandelbrot has used this in simulation of stock market fluctuations. What I have tried to do is to match the quality of the Prime fluctuation as seen in the sequence:
 

1) l(x)=(P(x)-P(x-1))/2

by a Mandelbrot cartoon used in a Bescovitch-Ursell function as found in Edgar's ``Measure, Topology and Fractal Geometry'' on page 202. In matching the l(x) to the first three prime differences as {1/2,1/2,1}, I have used a low ramp approach as only a vague approximation. There are two problems with this approach: first, the overall prime distribution in l(x) terms gradually increases while the multifractal only cycles; second, the dimension that is used is 5/8 from the Ingham 1937 proof and is thus not a modern one , but only a rational approximation. Even with these flaws the output is as good as some of the best Bonacci type sequences that I have tested in my Pickover number format. 

Treating the prime set evolution as a time series and using modern mechanics as developed by Dr. Mandelbrot and associates to simulate this natural sequence seemed like a logical next step. If I or one of the other fractalist didn't do it, I doubt that the number theory or physics people would get to it for several more years. To tell you the truth even with some success in this prime business, I'm getting sick of primes. It is such a limited one dimensional integer set! I hope that people will read this article and use this method which is a very practical more than theoretical one.

True Basic Program:

161 PRINT " used in a multifractal Prime sequence generator at the Ingham 5/8 dimension"

170 PRINT " by R. L. Bagula 11 Oct. 2001 (C)copy rights reserved"

180 LET a=2

190 LET s0=5/8

200 FOR w=0 to 2*pi step 2*pi/4500

210 LET sx=0

220 FOR k=1 to 20

230 LET sx=sx+a^(-s0*k)*bis(a^k*w)

240 REM Besicovitch- Ursell function summed

250 NEXT k

260 LET x=w

270 LET y=sx

280 LET x1=1026*x/(2*Pi)

281 LET c=c+1

282 REM scaled multifractal used in the P(x)=P(x-1)+2*l(x) form as the l(x) sequence

283 LET f(c)=f(c-1)+2*(1026*y/(2*Pi))/(375/81)

290 LET y1=750/2+750*y/(2*Pi)

300 SET COLOR 255

310 IF c< 650 then PRINT Int(f(c));" ";

311 IF c> 650 then PLOT x1,y1;

320 NEXT w

330 END

340 EXTERNAL

350 DEF bis(x)

351 REM Mandelbrot Multifractal cartoon function of three lines defined

352 REM 0<x2<x3<1

353 REM 0<y3<y2<1

354 REM unit square domain

355 REM lines continious

356 REM page 33 of "Multifractal and 1/f Noise"

360 LET x1=1/3

370 LET y1=1/32

380 LET x2=2/3

390 LET y2=1/31

400 LET x= mod(x,1)

410 IF x>=0 and x <= x1 then LET y=y1*x/x1

420 IF x>x1 and x<= x2 then LET y=(y2-y1)*x/(x2-x1)+y2-(y2-y1)*x2/(x2-x1)

430 IF x>x2 and x<=1 then LET y=(1-y2)*x/(1-x2)+1-(1-y2)/(1-x2)

440 LET bis=y

450 END DEF

Last Modified: by Roger L. Bagula 11 Oct. 2001©