Vampire numbers


Document created December 10 2002.
Sections "Vampire with 100025 fang pairs" and "Exponentially growing number of vampires" added June 27 2003.

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.

Vampire number counts
Digits Vampire
ratio

Vampires with at least f different fang pairs

Vampire
equations
Prime
vampires
f=1(all vampires) f=2 f=3 f=4 f=5
4 1/1286 7 0 0 0 0 7 0
6 1/6081  148 1 0 0 0  149 5
8 1/27881  3228 14 1 0 0  3243 57
10 1/82984 108454 172 0 0 0  108626 970
12 1/204980 4390670 2998 13 0 0 4393681 26653
14 1/431813 208423682 72630 140 3 1 208496456 923920

Vampire ratio is (n-digit vampire numbers)/(n-digit integers) and not a ratio of performed tests.
Vampire equations are all equations of the form vampire = fang1 · fang2, i.e. each vampire number counts for each different fang pair. Prime vampires obviously only have one fang pair.
The vampire equations for all table counts below 15 are on this page.

First 15 vampires with exactly 2 fang pairs
125460 = 204 · 615 = 246 · 510  11930170 = 1301 · 9170 = 1310 · 9107
12054060 = 2004 · 6015 = 2406 · 5010  12417993 = 1317 · 9429 = 1347 · 9219
12600324 = 2031 · 6204 = 3102 · 4062  12827650 = 1826 · 7025 = 2075 · 6182
13002462 = 2031 · 6402 = 3201 · 4062  22569480 = 2649 · 8520 = 4260 · 5298
23287176 = 2673 · 8712 = 3267 · 7128  26198073 = 2673 · 9801 = 3267 · 8019
26373600 = 3600 · 7326 = 3663 · 7200  26839800 = 2886 · 9300 = 3900 · 6882
46847920 = 4760 · 9842 = 6290 · 7448  61360780 = 7130 · 8606 = 7613 · 8060
1001795850 = 10170 · 98505 = 19701 · 50850


First 15 vampires with 3 fang pairs:
13078260 = 1620 · 8073 = 1863 · 7020 = 2070 · 6318
107650322640 = 140532 · 766020 = 153204 · 702660 = 200760 · 536214
113024597400 = 125100 · 903474 = 152100 · 743094 = 257400 · 439101
119634515208 = 195351 · 612408 = 234156 · 510918 = 285513 · 419016
134549287600 = 138650 · 970424 = 145700 · 923468 = 182900 · 735644
135173486250 = 164175 · 823350 = 328350 · 411675 = 361185 · 374250
138130447950 = 140415 · 983730 = 308913 · 447150 = 330891 · 417450
146083269717 = 167409 · 872613 = 204687 · 713691 = 237897 · 614061
150967233648 = 163548 · 923076 = 327096 · 461538 = 367983 · 410256
216315684000 = 316251 · 684000 = 351000 · 616284 = 421668 · 513000
221089445500 = 225500 · 980441 = 440198 · 502250 = 441980 · 500225
315987404670 = 348705 · 906174 = 446859 · 707130 = 453087 · 697410
463997983680 = 469938 · 987360 = 478380 · 969936 = 493680 · 939876
472812953760 = 629370 · 751248 = 657342 · 719280 = 671328 · 704295
10174695862032 = 1322058 · 7696104 = 1406019 · 7236528 = 1809132 · 5624076


First vampires with 4 and 5 fang pairs:
16758243290880 = 1982736 · 8452080 = 2123856 · 7890480 = 2751840 · 6089832 = 2817360 · 5948208
18762456533040 = 2558061 · 7334640 = 3261060 · 5753484 = 3587166 · 5230440 = 3637260 · 5158404
24959017348650 = 2947050 · 8469153 = 2949705 · 8461530 = 4125870 · 6049395
= 4129587 · 6043950 = 4230765 · 5899410

Smallest vampire number with 1, 2, 3, 4, 5 fang pairs:
1260, 125460, 13078260, 16758243290880, 24959017348650

Vampire with 100025 fang pairs
A 70-digit vampire number with 100025 different fang pairs:

1067781345046160692992979584215948335363056972783128881420721375504640

= 10678480810942174645657393025226495 · 99993750417382556182683103817340672
= 10678627541790580122508405533952248 · 99992376442330371917154164866387680
= 10678870645463554371658209457751760 · 99990100123533226388114347982829264
= 10678925660351864576350317228737136 · 99989585002034549234496184711872240
= 10679051531926761484527503920563120 · 99988406447319285718532648847633072
= 10679070628515510276363214421074560 · 99988227645479313532487533888609194
= 10679277844716332381108925372508482 · 99986287516103441645630395905467520
= 10679542601279227006934138728832640 · 99983808755841163513535641459770426
= 10679621271352355185449739854675480 · 99983072237818042661261004507343968
..... (100015 fang pairs omitted here)
= 32676369193453808819526186585907440 · 32677478294010514277129531489030256

The tested number was carefully chosen.
See choice of vampire, 1000 fang pairs or all fang pairs (3.2 MB zip file).

Search of small vampires
I wrote a very efficient C program to find vampire numbers: Algorithms and C source.
The 4390670 12-digit true vampire numbers were computed to a 128 MB text file in 9 minutes on my
Athlon XP 1500+ with 133 MHz ram on November 10 2002. As far as I know, Walter Schneider was first to compute them but a bug gave him too many numbers. We agree on the count now.

The 208423682 14-digit vampires were computed to a 7 GB file in 19 hours on November 12-13
2002.

Vampire patterns
Schneider writes that Fred Roushe and Douglas Rogers were first to find a pattern to generate infinitely many true vampires. He references an undated manuscript I have not seen. All known patterns fill 0's in the middle of one or both fangs. Such patterns are abundant and easy to find with a computer search. Extra conditions are required to make huge vampires interesting.

Weisstein ascribes this pattern to Roushe and Rogers:
10524208 = 2501 · 4208
1005240208 = 25001 · 40208
.....
1·102n+3+524·10n+1+208 = (25·10n+1) · (40·10n+208)
The formula produces vampires with 2n+4 digits, i.e. the 2 examples are for n=2 and n=3.
If the fangs are called x and y then the vampire equation can be written:
rev(x)·10n+2+y = x · y, where rev(x) is the decimal reverse of x.

I noticed that the smallest vampire with 2 fang pairs starts a pattern of vampires with 2 fang pairs:
125460 = 204 · 615 = 246 · 510
12054060 = 2004 · 6015 = 2406 · 5010
.....
12·102n+54·10n+60 = (2·10n+4) · (6·10n+15) = (2.4·10n+6) · (5·10n+10)

This formula generates squared vampires:
(94,892,254,79510n+1)2 = 9,004,540,020,079,200,492,025102n+189,784,509,59010n+1
The form of the fang makes it ideal for proving large primes. I have used PrimeForm/GW to find and prove the fang prime for n = 41, 65, 75, 257, 633, 730, 4755, 4780, 16868 and 45418.
The last prime has 45429 digits and was number 1301 on the list of the 5000 largest known primes when I submitted it:
                          (22 November 2002, 04:32pm)
1301a 94892254795*10^45418+1       45429 p97  2002
...
p97  Jens Kruse Andersen, PrimeForm
The corresponding vampire has 90858 digits.
Update September 9 2007: 9489225479510103294+1 is prime!
The vampire has 206610 digits. Many exponents between 45418 and 103294 have not been tested.

Exponentially growing number of vampires
The simplest vampire pattern is:
(3·10n) · (5·10n+1) = 15·102n+3·10n
The first cases are: 30·51=1530, 300·501=150300, ...
It remains a vampire when an arbitrary number of the digits "351" is inserted in the second fang, as long as there is at least one "0" to the left of every "3".
Example: 300000000000 · 500351035101 = 150105310530300000000000
Each "0351" in the second fang is multiplied by 3 and becomes "1053" in the vampire.
This pattern trivially shows that the number of d-digit vampires tends to infinite.
Moreover, it grows faster than any polynomial since sufficiently large n can give an arbitrary number of "free variables" (positions to place "351"). This is also called exponential growth.

Theorem: The number of d-digit vampires (d even) grows faster than db-1 for any given b.
Proof: Consider vampires of the above pattern with exactly b occurrences of "351". Split the second fang in b parts, one for each "351". If d is big enough, then each "351" can move independently in at least d/(10b) positions. The value 10 is not important, only that there is some constant. The number of combinations is at least (d/(10b))b. If b is fixed then this grows faster than db-1 which completes the proof.

A formula for the vampire sequence with only one 0 to the left of all 3's is:
(30·10n) · (50·10n+3510·(10n-1)/9999+1) = 15·102n+2+1053·(10n-1)/9999·10n+2+30·10n
Here n must be divisible by 4. 3510·(10n-1)/9999 is the number with n/4 concatenations of "3510", and 1053·(10n-1)/9999 is n/4 concatenations of "1053".

Gigantic vampire number
November 17 2002 I found the 10060-digit vampire number:
S · (S+12958410996), where S is 503 repetitions of "9514736028".
It is the smallest solution of form S · (S+9n) and required 12958410996/9+1 = 1,439,823,445 attempts to find. All digits occurs more than 1000 times in the decimal expansion.
The fangs were not generated by a vampire pattern and do not contain adjacent 0's. The old record was a 100-digit vampire found by Myles Hilliard March 9 1999. I broke that several times in the week before the above record.

Links about vampire numbers:
Clifford A. Pickover's original vampire number post from 1994
A 1999 post from Casey Billett ascribing the modulo 9 congruence to Pete Hartley
Eric Weisstein's World of Mathematics
John Child's Vampire Numbers (obsolete results)
Carlos Rivera's Prime Puzzles and Problems
Neil Sloane's The On-Line Encyclopedia of Integer Sequences (search on vampire numbers)
Henry Ernest Dudeney's Amusements in Mathematics at Project Gutenberg (Puzzle 85: The cab numbers)

This page was made by Jens Kruse Andersen and last updated 12 March 2012.
E-mail me with any comments or interesting results.jens.k.a@get2net.dk   home