The Museum of HP Calculators

Random Number Generators for the HP-41

This program is Copyright © 2004 by Jean-Marc Baillard and is used here by permission.

This program is supplied without representation or warranty of any kind. Jean-Marc Baillard and The Museum of HP Calculators therefore assume no responsibility and shall have no liability, consequential or otherwise, of any kind arising from the use of this program material or any part thereof.

Overview

-Several pseudo random number generators are listed on this web page:
-"RNG1" "RNG2" "RNG3" work on every HP-41.
-"RNG4" requires a Time-Module.
-Finally, the last program is an attempt to play ( win? ) the lottery...

Program#1

-A well known RNG is given by the formula:  xn+1 = FRC ( 9821 xn + 0.211327 ) which provides 1 million random numbers.
-The following program gives 1,000,000,000  random numbers r  ( 0 <= r < 1 ). The formula  xn+1 = FRC ( 98 xn + 0.236067977 ) is used.
-The coefficient 98 = 43,046,721 may be replaced by  a   where  a = 1 ( mod 20 )
and  0.236067977  -------------------    b   -----   b*109 is not divisible by 2 or 5.

01  LBL "RNG1"
02  9
03  ENTER^
04  ENTER^
05  R^
06  *
07  FRC
08  *
09  FRC
10  *
11  FRC
12  *
13  FRC
14  *
15  FRC
16  *
17  FRC
18  *
19  FRC
20  *
21  FRC
22  5
23  SQRT
24  +
25  FRC
26  END

( 35 bytes / SIZE 001 )

Example:  0.2   XEQ "RNG1"  yields  0.436067977
R/S                      0.779021394   ... etc ...

Program#2

-"RNG2" provides 9,999,999,996 random numbers with the formula:  xn+1 = ( 1059 xn ) MOD p   where p = 9,999,999,967 is the greatest prime < 1010
xn are integers between 0 and p ( exclusive ) which are then divided by p to be reduced to a number between 0 and 1.
-This routine works well because the MOD function gives exact results even when the operands are greater than 1010.
-Actually, the exponent 59 may be replaced by any integer m provided m is relatively prime to p-1 = 2*3*11*457*331543
but I don't know what is the best choice.
-Unlike "RNG1" and other routines based upon the same type of formulae, the least significant digits don't go through any cycle of ten, one hundred and so on.
-Register R00 is used to store the different  xn integers.

01  LBL "RNG2"
02  RCL 00
03   E59
04  *
05  10
06  10^X
07  33
08  -
09  MOD
10  STO 00
11  LASTX
12  /
13  END

( 26 bytes / SIZE 001 )

Example:   1  STO 00
XEQ "RNG2" gives  0.3129146797   ( and R00 = 3129146787 = 1059 ( mod p ) )
R/S                          0.6904570204    ( R00 = 6904570181 ) ... etc ...

-Actually if p is a prime,  ( Z/pZ-{0} ; * ) is a group
and if a is an integer,  the number of distinct elements in the subset   { 1 ; a ; a2 ; ....... ; ak ; .... } ( mod p )   divides  p-1
-If  p-1 is the smallest positive integer q such that aq = 1 ( mod p ) , then the sequence   a ; a2 ; ....... ; ak ; .... ; ap-1 ( mod p )
is a permutation of  1 ; 2 ; ...... ; p-1
-In particular, if  p = 2p' + 1  where p' is also a prime, and if ap' is not equal to 1 ( mod p )  then  a  satisfies the required property.
-For instance,  p = 7,841,296,787 = 2*3,920,648,393 + 1              7,841,296,787 and 3,920,648,393 are primes and  -1024 = 4,851,307,369  ( mod p )
satisfies  (-1024)p' = -1  therefore the routine:

E24
*
CHS
7841296787
MOD                  gives  7,841,296,786  random integers

-These ideas may be used to create your own RNG.

Program#3

-The following algorithm is given by Clifford Pickover in "Keys to Infinity" ( John Wiley & Sons )  ISBN 0-471-11857-5

01  LBL "RNG3"
02  LN
03   E2
04  *
05  1
06  MOD
07  END

( 17 bytes / SIZE 001 )

Example:  0.1   XEQ "RNG3" produces  0.74149070
R/S                                 0.09073404  ... etc ...

Program#4

01  LBL "RNG4"
02  DATE
03  TIME
04  +
05   E49
06  *
07  PI
08  MOD
09  LN1+X
10  R-D
11  FRC
12  END

( 25 bytes / SIZE 000 )

-I can't give any example since the result depends on the instant you press R/S

Winning the Lottery?

-If you need, for instance, 7 integers between 1 and 49 , you can use 7 random numbers between 0 and 1,
multiply them by 49 , add 1 and take the integer part of the result.
-The small routine hereafter is another possibility, if you accept to calculate the integer part in your mind:

01  LBL "\$\$\$"
02  R-D
03  49
04  MOD
05  1
06  +
07  END

( 16 bytes / SIZE 000 )

Example:         41  XEQ "\$\$\$"  yields   47.1270
R/S                   6.1759
R/S                 11.8535
R/S                 43.1567
R/S                 23.6963
R/S                 35.6962
R/S                 37.2418         suggesting     47-06-11-43-23-35-37

Notes:

1-I'm not a statistician and I can't assure all these RNGs would stand up to sophisticated tests,
but one may use his imagination in devising variations.
2-If you win one million dollars thanks to one of these programs, I accept to share the jackpot...