UPDATE: I am quite honored to state that this post has been endorsed by the man who defined Vampire Numbers, Clifford A. Pickover:

Tremble ghouls and boos: It’s a Spherical Cow Halloween! Those of you with timid hearts and fragile minds may not wish to continue. Hark dear reader; I beseech you, if you are adverse to maths macabre, chilling calculations, diabolic divisions, and fearful factoring, read no further. But if you are made of sterner stuff, and you reserve does not pall, tread onwards and I promise you delectably devilish mathematical diversions. For it is well-known by the old folk and those that care to listen that in these nights ‘fore All Hallows’ Eve, ghosts and goblins and bogeymen bewitch this site, whispering forbidden formulae to all fool enough to hear it.

Ah, so you’ve chosen to continue. Your fearlessness shows your true red blood, and here normally that would be a liability – You see we are in the presence of the nosferatu! Aye, the very children of the night! From your pallor, I see that you feel their presence in your very ichor, but your trepidations are unfounded. For, you see these parasites aren’t sustained by the instrument of your pulse; rather they feed off each other. Cannibal bloodsuckers, how horrid! But that is indeed what they are- The dreadful Vampire Numbers!

“What manner of demon is this,” you ask, and so I will tell you: Vampire Numbers were first described by the brave mathematical author Clifford A. Pickover in 1994. Risking life, limb, and sanity he valiantly scrawled this enigmatic message:

*If we are to believe best-selling novelist Anne Rice, vampires resemble humans in many respects, but live secret lives hidden among the rest of us mortals. Consider a numerical metaphor for vampires. I call numbers like 2187 vampires numbers because they’re formed when two progenitor numbers 27 and 81 are multiplied together (27*81 = 2187).*

*Note that the vampire, 2187, contains the same digits as both parents, except that these digits are subtly hidden, scrambled in some fashion. Similarly, 1435 is a vampire number because it contains the digits of the progenitors 35 and 4: (35*41=1435).** *

*These vampire numbers secretly inhabit our number system, but most have been undetected so far. I believe there are only six four-digit vampires in existence, but have no idea if there are any larger vampire numbers. *

* *

Mathematicians have a penchant for games, and a greater penchant for puns, so Mr. Pickover’s modest message spawned some monstrous results. A legion of vampire hunters arose. Their numbers and tenacity would shame even the likes of the Van Helsings and Belmonts. Mathematicians and computer scientists, amateur and professional, jumped on the challenge, and what they found lurking in the integers was more nightmarish than any could have predicted.

Poor Pickover could never have known the multitude of fiends that would be released and the magnitude of their monstrousness when he made his modest post, but they are unleashed now. For example, he was wrong when he predicted that there are only six four-digit vampire numbers, in fact, there are seven. I will show them here, so that you may be better able to protect yourself:

21 x 60 = 1,260

15 x 93 = 1,395

35 x 41 = 1,435

30 x 51 = 1,530

21 x 87 = 1,827

27 x 81 = 2,187

80 x 86 = 6,880

Worse yet, they do come bigger, much bigger. Here is a wicked, six digit vampire:

204 x 615 = 125,460

As they were hunted, our taxonomy improved, and what Pickover had termed *progenitors* we now call *fangs*. So, for instance, in the eight digit behemoth:

1,301 x 9,170 = 11,930,170

1,301 and 9,170 are the vampire’s fangs. To summarize, so that you may know how to identify the devils: A Vampire Number is a number that contains every digit of the two numbers multiplied together to create it. More technically, A Vampire Number, *v*, is a number with an even number *n* digits that is the product of a pair of numbers, x,y, of *n/2* digits each such that appearing in *v* are all the digits of *x* and *y* in any order. To avoid trivialities, a Vampire Number is not allowed to have trailing zeros. Thus, 350 x 410 = 143,500 is not considered a Vampire Number.

The search for Vampire Numbers has led to the discovery of some truly awesome nightmares such as this 100 digit vampire number found on March 9, 1999 by Myles Hilliard:

98765432109876543210987654321098765432108990776898 x 98765432109876543210987654321098765432109967196255 =

97546105798506325258725803993760097546163976172239484889869060971840117345

46933285857249127286116990

However, a word of caution to those who would hunt the unclean themselves, finding ever larger monsters has become a quest, nay an obsession for some, and the world record is currently hard to top. Jens K. Anderson, a braver man than most, currently has the distinction of staring farthest into the abyss. He has seen the beast and lived to tell about it. At 10,060 digits, he has earned his fame. Go here to see the goliath in its lair, but I cannot be held responsible for the consequences to your sanity or your soul.

Anderson’s number is literally indescribably huge. There is not currently a word to classify a 10,060 digit number. It is larger than anything the physical world can offer- the number of atoms in the universe is estimated to be 10^80, an 81 digit number. Conventional naming systems offer a name that is more cumbersome than 10,060 digits, and therefor I propose a new name for numbers that are 10,060 digits. My suggestions are either Vampillion or Vampirillion. Which do you prefer?

** ****Addendum: **A heroic reader, Mr. Rossi, observed that I had neglected to properly arm you against these numbers. I have ruminated over this quandary and have devised the following guide for dealing with these abominations. For the benefit of all mankind I preserve here this historic exchange:

**Mr. Rossi : **

*First you charm us with a Moobius Mooroboros and then you do this?! I was not prepared for this kind of sucker punch. Not only that but you only mentioned how to identify these horrific integers. What must we do to protect ourselves and our loved ones?*

**My Response:***Traveler, your admonition is well deserved and it is an error I fear may have cost the life and soul of too many a mortal. But you see the wards against the foul things are well known and widely written. Recall Van Helsing’s words (with which I know you are familiar): “I shall cut off her head and fill her mouth with garlic, and I shall drive a stake through her body.”*

*This shall be our stratagem. Consider the wretched 13,078,260 = 1620 × 8073. First, cut off its head. Now, the Vitruvian proportions tell us that a man’s head is one eighth its height, so from the beast we slice off its first eighth: 13,078,260 becomes 3,078,260. Then we are to fill its mouth with garlic. Garlic is related to the astrological sign ares, which is numerological related to the number nine. So we jam 9 into its gullet: 3,078,260 becomes 39,078,260. Finally we drive a stake through its heart. Take firm hold of one (1) and nail it into its chest: 39,078,260 becomes 390,718,260. Now the beast is good and dead.*

* *This is the first in a series of Halloween posts. If you enjoyed it, you may like the others: Infernal Integers*, *Zombi(nacci), The Sum of All Fears

First you charm us with a Moobius Mooroboros and then you do this?! I was not prepared for this kind of sucker punch. Not only that but you only mentioned how to identify these horrific integers. What must we do to protect ourselves and loved ones?

Traveler, your admonition is well deserved and it is an error I fear may have cost the life and soul of too many a mortal. But you see the wards against the foul things are well known and widely written. Recall Van Helsing’s words (with which I know you are familiar): “I shall cut off her head and fill her mouth with garlic, and I shall drive a stake through her body.”

This shall be our stratagem. Consider the wretched 13,078,260 = 1620 × 8073. First, cut off its head. Now, the Vitruvian proportions tell us that a man’s head is one eighth its height, so from the beast we slice off its first eighth: 13,078,260 becomes 3,078,260. Then we are to fill its mouth with garlic. Garlic is related to the astrological sign ares, which is numerological related to the number nine. So we jam 9 into its gullet: 3,078,260 becomes 39,078,260. Finally we drive a stake through its heart. Take firm hold of one (1) and nail it into its chest: 39,078,260 becomes 390,718,260. Now the beast is good and dead.

Pingback: Infernal Integers | Spherical Cow

Pingback: Zombi(nacci) | Spherical Cow

Pingback: 2010 in review | Spherical Cow