Program PERFECT and PRIME By 4md (Graham Gallagher) Use Experiment with numbers This program was inspired by a question from my son Ian (aged 11). He'd been reading the section on MATHEMATICS in the GUINNESS BOOK OF RECORDS and was intrigued by perfect numbers. I had to be honest and confess that I didn't know what these were. The explanation followed that a number is "perfect" if all of it's factors, other than itself, add up to equal the number: ie. 1+2+3 = 6, the lowest perfect no. The next is 1+2+4+7+14 = 28 These were both known to my son. What he asked was did I know of any others? I obviously didn't but suggested that I could write a short program to search them out. Hence a fore-runner of the program PERFECT was produced which sluggishly found the next perfect number and after a few incremental restarts over 2 or 3 days found the next. After the above exercise I decided to read the bit in GUINESS BOOK OF RECORDS for myself and discovered that to dated there are only 32 known perfect numbers; the highest being: ((2^756839)-1)*(2^756838) This number if printed out in full has 455663 digits and would fill over 100 sheets of A4 paper. The expression on the left hand side of the multiplier is the largest known prime number and was discovered in February 1992. Armed with this information I thought that I could reduce the search time by using a sub-routine which identifies prime numbers. Unfortunately, I've not been able to make use of such a sub-routine in the program searching for perfect numbers. Hence I've included it as a stand alone program (ie.PRIME). This simply identifies a screen-full of prime numbers and then stops. A more useful approach was to assume that all perfect numbers follow the same expression as the highest known perfect number: ie. perfect no. A=(2^J)*((2^(J-1))-1) The latest program (ie.PERFECT) therefore produces a value for "A" and then importantly checks to see if the number is in fact "perfect". This soon takes the BEEB quite some time to do, but fortunately the numbers found do agree with those calculated by my first version of the program which laboriously checked each number from one upwards. I'll now point out that it takes the BEEB about 90 seconds to find the fourth perfect number using the program PERFECT as it now stands. I have to admit that I haven't left it running long enough to find the fifth one. However, I asked a work colleague to run the program on a PC486 machine and managed to calculate this fifth perfect number with some certainty. Some further refinements have been made in order to deal with the very large numbers involved and the PC486 has now regurtitated what we believe to be 8 out of the 32 known perfect numbers. We are not entirely sure that they are correct and there may be other perfect numbers lower than the ones we've found which don't fit the expression used in the program. If anyone has a list or knows of a source of reference I'd appreciate the assistance (Tel. 0788 823214). Finally, perhaps someone would care to convert the program to machine code. From: K4D (Lorin Knight) Programs: MAGIC3D, MODE4PR Associated files: MEX-HAT, FURROWS Use: Creating "magic" 3-D illusions You may have seen the MAGIC EYE books or posters in which a 3-D image mysteriously appears from what, at first sight, appears to be a random pattern of coloured dots or shapes. MAGIC3D enables simple black-and-white pictures of this type to be produced on a BBC computer. A MODE-4 picture is built up on the monitor screen, a process typically taking a little over an hour, and when completed is dumped onto disk. In order to see the 3-D effect you need to print it out. This can be done using the associated program MODE4PR If you have an Epson 9-wire dot matrix (or compatible) printer. Otherwise I'm afraid you will have to find some other way to print it. Two sample screen dumps, MEX-HAT and FURROWS are included on the drive 2 side of this disk so that you can print out examples showing what the program can produce.
EDITOR..... ADFS version the files are
in DIR $.Software
The recommended way of seeing the 3-D illusion is to hold the printout right up to the nose and to stare straight ahead, while moving the picture away very very slowly. The illusion should then gradually appear. You may not succeed first time but you will soon get the knack after a litle practice. Don't try doing this with the display on the monitor. It could be harmful to put your face rightup to the screen . In MAGIC3D a mathematical formula (in line 80) is used to define the 3-D image. As written it produces a pyramid. Within the REM statements at the end of the program you will find alternative lines which can be used to produce other images. HOW IS THE 3-D ILLUSION PRODUCED? For each line of the picture the program produces a different 60-pixel loop of random dots and spaces (1s and 0s). To produce a background plane, the loop is repeated throughout the line, making a pattern which repeats itself five times across the picture. When you are "tuned in" to the 3-D mode, instead of both eyes focussing on the same pattern, they focus separately onto two adjacent patterns. The brain is then fooled into seeing a single pattern which is some distance behind the plane of the paper. To produce a foreground plane, the program reduces the loop length to 45. This brings the repeating patterns closr together and the brain perceives the image to be nearer the paper. By varying the length of the loop between 45 and 60, one can choose any of 16 different depths for the image. Surprisingly, it turns out that it is quite simple to mould a 3-D shape from these 16 depths, just by suitably varying the length of the loop along each line of the picture. All one has to do is to specify the required depth, and hence loop length, for each pixel position in the picture. If the loop is required to be x digits shorter, the next x digits are removed from the loop and the resultant gap closed up. If the loop is required to be x digits longer, a gap is opened up and an extra x random digits inserted. IMAGE DESIGN The formula used for defining the image calculates the depth, D%, for the screen position, X% Y%. D% should be between the limits, 0 (foreground) to 15 (background), corresponding to loop lengths from 45 to 60. When we define the loop length at any particular spot in the picture, we are really defining the position which will be seen by the right eye. The corresponding position seen by the left eye will be 45 to 60 pixels to the left and the resultant position perceived by the brain will be midway, i.e. between 22 and 30 pixels to the left. I have found that an adequate, although not perfect, correction for this can be made by assuming the centre line of the picture to be at X%=740 instead of X%=640. NOTES ON USING THE PROGRAMS Before using MAGIC3D both programs should be copied onto a separate blank disk - so that there is room for screen dumps and any amended versions of MAGIC3D that you may want to experiment with. This is not essential however if you just want to take printed copies from the two sample screen dumps. The first line of each program is *RAMOFF. This is essential if you have a shadow RAM. Otherwise the line should be deleted to prevent it producing an error message.
EDITOR........ For compatibility I have
added line 5 to both programs. This
line runs the program from line 20 if
the *ROMOFF instruction is not
recognised by your machine.
The screen dump produced by MAGIC3D is always called PICTURE. You will need to rename each dump before producing another - otherwise you will find it overwritten!
With suitable modifications MAGIC3D can be made to work in MODE 1 and produce a pattern in four colours instead of just black and white. I havenot pursued this possibility because I don't have a colour printer. Alternatively it can be modified to produce a MODE-0 picture. There will then be twice as many pixels in each line and the loop length can vary from 90 to 120, giving 31 graduations of depth. The resultant image is smoother but, to my mind, loses some of the impact of the coarser MODE-4 picture - and the time taken to compose the picture is much longer. Although it is easy to define a mathematically derived image it is much more difficult to define, say, a dinosaur or an elephant. Conceivably such an image could be defined using a MODE-2 picture in which each of the 16 possible colours represented a different depth - but I don't think I shall be attempting this! To: 999 (all members) From: K2K (Peter Davy) Subject: PALINDROMIC NUMBERS I could not resist K3X's challenge in the no.34 issue to write a program to find palindromic numbers whose squares are also palindromic. Elsewhere on this disk you will find the program LAPPY which is my final solution to this problem. You should now find LAPPY and produce a listing on the printer to look at as you read this article. It is evident from the example quoted by K3X (101010101 squared equals 10203040504030201) that we need to go well beyond the normal arithmetic capability of the Beeb i.e. 9 significant figures. I remembered having read somewhere about a program which enables arithmetic to be performed on string representations of long numbers. A search through my quite large collection of BBC micro books eventually turned up Utility 31: Multiple Precision Arithmetic in the book "INVALUABLE UTILITIES for the BBC MICRO" by Jeff Aughton, published by PAN in 1984. Utility 31 takes the form of a procedure PROCfermat which calls up several other procedures. The complete utility enables addition, subtraction, multiplication and division. In the version which appears at the end of my program LAPPY I have only included the parts that are needed for multiplication. If two numbers are assigned to A$ and B$, for example A$="98765432123456789" and B$="12345678987654321", PROCfermat will produce the product of the two numbers as C$. For PROCfermat to work the arrays A%( ), B%( ) and C%( ) must be dimensioned 255. This is done at line no.15. My first attempt was to start with the smallest two digit number, 10 and to repeatedly add one to it. Each answer was turned back to front, i.e. reversed and then compared with the original. If the two were the same a palindromic number had been generated, and it was squared and the same operations applied to see if the square was palindromic. This program worked all right but it was far too slow. I had set my sights on going up to at least 999999999 for the numbers being squared. But 999999999 is a very big number. 999999999 seconds is more than 31 years! Even if numbers are generated at the rate of 10 to the second it would take 3 years to get from 10 to 999999999. I decided to try to generate the palindromes for squaring, directly. For example the smallest 6 digit palindrome is 100001 which could be made by joining together 100 and 100-reversed i.e. joining 100 and 001. The next six digit palindrome is 101101 made by joining 101 and 101. Then comes 102201 by joining 102 and 201. Now the program only needs to add 1 repeatedly to 100 until 999 is reached to generate all the six digit palindromes from 100001 to 999999. For numbers with an odd number of digits things are slightly different. For example the smallest 7 digit palindrome is 1000001. This and subsequent palindromes can be built up as follows: 1000001 100 + 0 + 001 1001001 100 + 1 + 001 1002001 100 + 2 + 001 and so on to: 1009001 100 + 9 + 001 1010101 101 + 0 + 101 1011101 101 + 1 + 101 1012101 101 + 2 + 101 and so on to: 1019101 101 + 9 + 101 1020201 102 + 0 + 201 and so on to: 9999999 999 + 9 + 999 Testing a number to see if it is a palindrome by reversing it and then comparing with the original is slow. In PROCtest I have used a quicker way of comparing the first and last digits and then the second and last but one and then the third and last but two and so on but stopping as soon as a mis-match is found indicating that the number is not palindromic. The values of 2 and 9 in line 20 means that all palindromes in the range of 2 digits to 9 digits will be generated i.e. from 11 to 999999999. Before running the program, press CTRL-N to put into paged mode. Then type RUN and press RETURN. All the palindromes generated will scroll past on the screen and those which have palindromic squares are sent to the printer. To keep the program running you will need a weight to hold down one of the SHIFT keys. To stop the output to examine it more closely, just take the weight off. It took my Master-128 about 19 hours to generate all the palindromes between 11 and 99999999 and 67 of them have palindromic squares. The print-out is interesting but I will say no more about it. You can run the program for yourself. If you would like to put PROCfermat through its paces, RUN the program for a short time then press ESCAPE. Now type A$="12345678987654321" and press RETURN and type B$="98765432123456789" and press RETURN. Any other big numbers will work just as well. Now type PROCfermat:PRINT C$ and press RETURN. The product of those two big numbers will appear on the screen. My thanks to K3X for raising this matter. What's the betting that someone has sent in six lines of code which does the job in five minutes?
Editor......... The program if run from
the 8BS menu does not produce anything
readable on the screen, so I have added
a MODE 7 instruction at the start of
"Lappy". When running programs such as
this, I use a piece of sellotape stuck
to the SHIFT key, pulled taut and
pressed down on to the front edge of
the computer. This holds the key down
much better than a coffee mug.