吸血鬼数字

在数学里有一种吸血鬼数，它是两个“尖牙数”的乘积，同时又是由这两个“尖牙数”的数字重新排列后得到的，如：27 * 81 = 2187, 35 * 41 = 1435 等。

1994年柯利弗德·皮寇弗在Usenet社群sci.math的文章中首度提出吸血鬼数。后来皮寇弗将吸血鬼数写入他的书Keys to Infinity的第30章。

吸血鬼数还要有一些条件：要有偶数个位数，每个“尖牙数”的位数是它的一半，两个“尖牙数”的个位不能同时为0。

约翰.查尔茨(John Childs)找到了一个40位的吸血鬼数，他的结果是：

98,765,432,198,765,432,198 * 98,765,432,198,830,604,534

= 9,754,610,597,415,368,368,844,499,268,390,128,385,732

The vampire numbers were introduced by Clifford A. Pickover in 1994.
(H. E. Dudeney's book "Amusements in Mathematics" from 1917 contained a variant in a puzzle called "The cab numbers")

Definitions:
A vampire number is a number which can be written as a product of two numbers (called fangs), containing the same digits the same number of times as the vampire number. Example:
          1827000 = 210 · 8700
A true vampire number is a vampire number which can be written with two fangs having the same number of digits and not both ending in 0. Example:
          1827 = 21 · 87
All vampire numbers (or just vampires) on the rest of this page are implicitly true. They must clearly have an even number of digits.
A prime vampire number (introduced by Carlos Rivera in 2002) is a true vampire number where the fangs are the prime factors.

The 7 vampires with 4 digits:
1260=21 · 60, 1395=15 · 93, 1435=35 · 41, 1530=30 · 51, 1827=21 · 87, 2187=27 · 81, 6880=80 · 86 

The 5 prime vampires with 6 digits:
117067 = 167 · 701, 124483 = 281 · 443, 146137 = 317 · 461, 371893 = 383 · 971, 536539 = 563 · 953

Modulo 9 congruence

An important theoretical result found by Pete Hartley:
          If x·y is a vampire number then x·y == x+y (mod 9)
Proof:
Let mod be the binary modulo operator and d(x) the sum of the decimal digits of x.
It is well-known that d(x) mod 9 = x mod 9, for all x.
Assume x·y is a vampire. Then it contains the same digits as x and y, and in particular d(x·y) = d(x)+d(y). This leads to:
          (x·y) mod 9 = d(x·y) mod 9 = (d(x)+d(y)) mod 9 = (d(x) mod 9 + d(y) mod 9) mod 9
            = (x mod 9 + y mod 9) mod 9 = (x+y) mod 9

The solutions to the congruence are (x mod 9, y mod 9) in {(0,0), (2,2), (3,6), (5,8), (6,3), (8,5)}
Only these cases (6 out of 81) have to be tested in a vampire search based on testing x·y for different values of x and y.