Vampire Numbers! Part 2

How many vampire numbers are there?
How long does it take to find them?
How often do you find them?
How fast does the number of numbers to search through increase?

Thinking about the topic immediately raises a lot of questions. Here is a table that answers some of these questions.

Vampire Numbers - John Childs
Digits in vampire number	Numbers in range (i.e. 10 to 99 = 90 numbers	Number of multiplications to be performed	Number of vampire numbers	Per cent of total	Multiplication rate (#'s/hr)*	Time to search all possibilities	Time increase factor
4	90	4,095	7	0.2	29,000,000	0.5 sec	.
6	900	405,450	148	0.04	16,000,000	1.5 min	180x
8	9,000	40,504,500	3243	0.008	14,000,000	2.5 hours	100x
10	90,000	4,050,045,000	108,577	0.003	6,500,000	26 days	250x
12	900,000	405,000,450,000	?	?	5,500,000	8.4 years	120x
14	9,000,000	4.05 E+13	.	.	3,600,000	1,3000 years	150x
16	90,000,000	4.05 E+15	.	.	.	.	.
18	900,000,000	4.05 E +17	.	.	.	.	.
20	9,000,000,000	4.05 E +19	.	.	2,000,000	2.3 billion years	.
22	9 E +10	4.05 E +21	.	.	.	.	.
24	9 E +11	.	.	.	.	.	.
26	9 E +12	.	.	.	.	.	.
28	9 E +13	.	.	.	.	.	.
30	9 E + 14	.	.	.	.	.	.
32	9 E + 15	.	.	.	.	.	.
34	9 E + 16	.	.	.	.	.	.
36	9 E + 17	.	.	.	.	.	.
38	9 E + 18	.	.	.	.	.	.
40	9 E + 19	.	.	.	600,000	1 E +30 years	.
50	9 E +24	.	.	.	365,000	2 E +40 years	.
60	9 E +29	.	.	.	300,000	2 E +50 years	.
100	9 E +49	.	.	.	100,000	5 E +90 years	.

Notice again, that there is a big drop in the multiplication rate when we move to 10 digit products. At this point, we are using numbers up to 9,999,999,999 which exceeds the computer's integer capacity entirely. At this point, the computer can no longer supply us with numeric variables to do the job, and we have to do it manually. It's not often that the computer actually lets us down! But we really have exceeded the computer's ability to do the arithmetic in which we are interested. At this point, all our nubmers are actually arrays of numbers. A 10, 12 or more digit number is stored with digits, each in its own cell, in an array with 10 or 12 elements.
Such an array can be visualized like so:

Num1( )

8

4

5

2

8

9

0

3

5

1

7

3

The QuickBASIC code to display such a number would look like this:

FOR N = 1 to 12 PRINT Num1(N); NEXT N

To add the results of the individual multiplications,we need a two dimensional array to store the intermediate results. You can picture the multiplication job as shown: Using arrays dimensioned up to large values, we can now work with numbes up to a size where our next restriction would be the computers RAM! And we could exceed that by writing our arrays to disk files. But we are getting ahead of ourselves. :-) A final array can hold the sum of the vertical columns of the two dimensional array, just as you would do if you did the calculation by hand. Don't forget another array to store the results of whether or not there is a carry and what its value is!

Num1( ) 1 2 3 4 5 6 6 5 4 3 2 1 Num2( ) 9 1 2 3 4 5 9 1 8 1 7 7 . . . . . . . . . . . . 8 6 4 1 9 6 5 8 0 2 4 7 . . . . . . . . . . . 8 6 4 1 9 6 5 8 0 2 4 7 0 . . . . . . . . . . 1 2 3 4 5 6 6 5 4 3 2 1 0 0 . . . . . . . . . 9 8 7 6 5 3 2 3 4 5 6 8 0 0 0 . . . . . . . . 1 2 3 4 5 6 6 5 4 3 2 1 0 0 0 0 . . . . . . 1 1 1 1 1 0 9 8 8 8 8 8 9 0 0 0 0 0 . . . . . . 6 1 7 2 8 3 2 7 1 6 0 5 0 0 0 0 0 0 . . . . . 4 9 3 8 2 6 6 1 7 2 8 4 0 0 0 0 0 0 0 . . . . 3 7 0 3 6 9 9 6 2 9 6 3 0 0 0 0 0 0 0 0 . . . 2 4 6 9 1 3 3 0 8 6 4 2 0 0 0 0 0 0 0 0 0 . . 1 2 3 4 5 6 6 5 4 3 2 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 9 8 8 8 8 8 9 0 0 0 0 0 0 0 0 0 0 0 1 1 2 6 3 5 1 7 4 6 4 1 5 5 3 2 3 9 4 9 2 8 1 7

Vampire Numbers - Information Summary

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