Ivo Dntsch               | Phone:  (++44) 1232 - 368 976
        School of Information and | Fax:    (++44) 1232 - 366 803
        Software Engineering      | E-mail: I.Duentsch@ulst.ac.uk
        University of Ulster      | http://www.infj.ulst.ac.uk/~cccz23/
        Newtownabbey, BT37 0QB

        N. Ireland

        March 16, 1995

        Help: In the course of my move to Belfast I have lost the file
        containing the registered users. Please, register again, if you
        want to be informed of new versions.
        ___________________________________________________________

        Changes in version 202b, Jan 13, 1992

           1. Several bugs have been removed.

           2. The array of the first thousand primes is now part
              of the data area.

           3. Some improvements have been made to the Pollard-Brent
              algorithm:

                a. It is now possible to set the starting value, the
                   shift, and the period when the gcd is taken.

                b. The shift is increased when the algorithm is not
                   successful. 

           4. The corresponding changes have been made to NUMTYPE.PAS
              NUMBERS.CFG, LFACTOR.OBJ, LPRIME.OBJ.

              Caution: These routines will not work with 202A, neither
              will these routines of 202A work with 202B.

           5. Registration procedure has changed; the program now may
              be freely used for educational purposes, under the
              condition that the use is registered with the author
              (Jun 13, 1992).

           Fixed January 20, 1992:

           The ESC key in the Arithmetic procedure of NUMBERS was not
           returned correctly.

           Fixed January 21, 1992:

           The n-th residue test was not working correctly for an even
           modulus. Changed routine to only test for quadratic residue.

           Removed option to set the number of primes to use.

           Updated the manual.

           Changed January 23, 1992

           Include compiled readme and manual.

        Changes in version 202c (June 10, 1992)

           Removed an inconvenience in ESC routine while factoring.

           Removed an error in the input routine for long numbers.

           Added support for trapping rings during the computation
           of aliquot chains.

           Streamlined computation of aliquot chains.

           Fixed June 28, 1992: File I/O error in saving aliquot chains

           Updated the manual files, July 02, 1992

           Fixed July 06, 1992: A bug in the division routine which in rare
           circumstances might cause the program to go into an infinite loop.

        Changes in version 202d, March 16, 1995

           Jan Pedersen discovered and fixed a bug in the factoring routine
           which on very rare occasions caused the program to go into an
           infinite loop.

           Changed address information
           Re-wrote manual in TeX
           Changed version number

        ___________________________________________________________

        Welcome to the NUMBERS calculator. It is designed to facilitate
        the computation of number theoretic functions for positive
        integers of up to 150 decimal digits on almost any PC.

        My main objective in writing NUMBERS was to allow students in
        a course in elementary number theory to do nontrivial examples
        and pursue investigative projects, and it has served quite well
        in this respect.

        The NUMBERS package is not in the public domain. It may be
        freely used for educational purposes and I recommend that
        you register its use with me. Registered users will be notified
        of new versions and will receive some support. To register, just
        send me an e-mail message.

        I should appreciate any reports on the suitability of the system
        for the teaching and learning of number theory; bug reports are
        also welcome, that is, if there are any bugs.

        The documentation can be found in NUMBERS.DOC, a PostScript
        version in NUMBERS.PS, and an on-line manual in manual.exe
        (Thanks to DRC).

        Disclaimer:

        Extensive care has been taken to assure that the program runs
        properly; however, I disclaim all reliability to the user for
        any direct, indirect or consequential loss arising from the use
        of, or inability to use, this program.

        No warranty is given that the program will work under all
        circumstances, or that it does what it claims it does.

        The NUMBERS package contains the following files:

        1.      The Calculator

        numbers.exe             ;the calculator
        numbers.hlp             ;calculator help

        2.      Documentation

        numbers.dvi             ;documentation, dvi
        numbers.ps              ;documentation, postscript
        read.me                 ;introduction

        3.      Turbo Pascal Interface

        numtype.pas             ;type definitions and link commands
        primes.inc              ;table of first 1000 primes

        4.      Object Files

        cmpre.obj               ;compare two long numbers
        ladd.obj                ;add, subtract
        lcoprime.obj            ;decide if n,m are coprime
        ldiv.obj                ;divide
        lfactor.obj             ;factor a number
        lgcd.obj                ;find greatest common divisor of n and m
        llcm.obj                ;find lowest common multiple of n and m
        lmp.obj                 ;power to a modulus
        lmult.obj               ;multiply
        lphi.obj                ;Euler's Phi function
        lprime.obj              ;decide if n is prime,related procedures
        numtostr.obj            ;convert long number to string
        lroot.obj               ;integral square root
        sigma.obj               ;sum of divisors
        strtonum.obj            ;convert string to longnum