Abstract: The nonlinear Cantor set that gives us the Prime numbers is not alone in nature. In this article I introduce several new invented functions that are based on the same kind of variation as the primes. New Chaos is involved in the way these new Pickover numbers evolve. 

Body: When is a failure not a failure? When in attempting to make a function that gives the primes you fail by inventing a new kind of mathematics for functions. Dr. Pickover wrote an article several years back about the sound that Batrachion sequences make.

A friend Ado ( Simone Caramel) who runs the egroup I help moderate came up with a way to make the primes using a Fibonacci sequence. These two facts along with my formula:
 

1) p(x)=p(x-1)+2*l(x)

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

gave me the idea that this difference formula was some kind of integer sequence of a

type like:
 

3) l(x)=a1*l(x-1)+a2*l(x-2)+a3*l(x-3)+....+an*l(x-n)

4) q(n)=q(n-q(n-1))+q(n-q(n-2))

With some experimentation I came up with a sequence of numbers that varies very much like l(x):
 

5) l1(x)=q(x)-x/2

which gives the Hofstadter integers:
 

6) r(x)=r(x-1)+2*(q(x)-x/2)

The variation of this new kind of sequence of integers is much like that of the primes: the plot of one against the other gives a noisy line:
 

7) (xa,ya)=(p(x),r(x))

The plot of the ratio:
 

8) R(x)=p(x)/r(x)

is a chaotic function. If the results ended just here, this would be new mathematics, but they don't: it also gives a pseudozeta function:
 

9) pz(s)=Product[1/(1-1/r(n)^s),{n,1,Infinity}]

such that another sequence of integers is necessary:
 

10) pz(s)=Sum[1/Sq(n)^s,{n,1,Infinity}]

This new sequence of integers is one that is very hard to solve for. I don't think that the Hofstadter is alone in giving this kind of function. Let's look at Ado's function:
 

11) f(x)=f(x-1)+f(x-2)

12) A(x)=A(x-1)+2*(f(x)-x)

This sequence is also a new function that will give a zeta like function and another sequence of integers that is very hard to solve for!
 

13) pzA(s)=Product[1/(1-1/A(n)^s),{n,1,Infinity}]=Sum[1/SqA(n)^s,{n,1,Infinity}]

Each of these functions has a set of zeros that like the Riemann zeta function zeros are indications of how the sets vary. I'm going to call this new type of number ``Pickover numbers'', because he was the first one to realize that a unique type of chaotic was involved in these sequences. These functions can be made up in varying ways from integer sequences that involve the Fibonacci like mechanics in a New Chaos way. 

True Basic Program:

60 REM HOFSTADTER'S BATRACHION SEQUENCE

70 DIM q0(0 to 4000)

80 LET q0(0)=1

90 LET q0(1)=1

91 FOR n= 2 to 4000

92 LET q0(n)=q0(n-q0(n-1))+q0(n-q0(n-2))

93 NEXT n

94

100 SET MODE "color"

110 SET WINDOW 0,1024,0,750

111 SET COLOR MIX(0) 1,1,1

112 SET COLOR MIX(1) 0,0,0

120 REM sieve

121 PRINT "running sieve takes time"

130 LET E=640*7

140 DIM P(0 to 4000)

150 LET Q=0

160 FOR N=1 To 8*E

170 IF N<4 Then

172 LET Q=Q+1

174 LET P(Q)=N

178 GOTO 280

179 END IF

180 LET I=0

190 LET T=2

200 LET J=Int(N/T)

210 LET K=J*T

220 IF N=K Then GOTO 280

230 LET I=I+1

240 LET L=T*T

250 IF L>N Then

252 LET Q=Q+1

253 IF q>e then GOTO 300

254 LET P(Q)=N

258 GOTO 280

259 END IF

260 LET T=I*2+1

270 GOTO 200

280 IF Q>E Then GOTO 300

290 NEXT N

300 CLEAR

301 PRINT" Primes and Hofstadter Integers"

302 Print" by Roger L. Bagula 7 Oct 2001 (C)"

303 DIM r(0 to 4000)

304 LET r(0)=1

305 rem Primes patterned as p(x)=p(x-1)+2*l(x)

306 rem l(x)=(p(x)-p(x-1))/2

307 rem where q(n)=q(n-q(n-1))+q(n-q(n-2))

308 rem for the Hofstadter integers is used to simulate l(n)

310 FOR x=1 to 1024

311 rem Hofstadter Batrachion integers : patterned after the prime numbers

312 LET r(x)=r(x-1)+2*(q0(x)-x/2)

313 SET COLOR 255

314 PLOT P(x)/11, r(x)/11;

315 

380 NEXT x

390 

460 END
 
 

Last Modified: by Roger L. Bagula 8 Oct 2001©