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Vampire Numbers

Vampire Numbers are a fascinating class of numbers in recreational mathematics where a number can be factored into two smaller numbers (called "fangs") that contain precisely the same digits as the original number.

Definition

A vampire number is a composite natural number with an even number of digits, that can be factored into two natural numbers (the fangs) such that:

  1. Each fang has half as many digits as the vampire number

  2. The two fangs together contain precisely the same digits as the vampire number

  3. At most one of the fangs can have trailing zeros

Mathematical Notation

For a vampire number \(v\) with fangs \(x\) and \(y\):

  • \(v = x \times y\)

  • \(v\) has \(2n\) digits, \(x\) and \(y\) each have \(n\) digits

  • The digits of \(v\) are a rearrangement of the combined digits of \(x\) and \(y\)

Examples

4-digit Vampire Numbers

  • 1260 = 21 × 60 (digits: 1,2,6,0 = 2,1 + 6,0)

  • 1395 = 15 × 93 (digits: 1,3,9,5 = 1,5 + 9,3)

  • 1435 = 35 × 41 (digits: 1,4,3,5 = 3,5 + 4,1)

  • 1530 = 30 × 51 (digits: 1,5,3,0 = 3,0 + 5,1)

6-digit Vampire Numbers

  • 125500 = 251 × 500 (digits: 1,2,5,5,0,0 = 2,5,1 + 5,0,0)

  • 126027 = 201 × 627 (digits: 1,2,6,0,2,7 = 2,0,1 + 6,2,7)

Properties

Rarity

  • There are 7 four-digit vampire numbers

  • There are 148 six-digit vampire numbers

  • There are 3,228 eight-digit vampire numbers

  • The count grows roughly exponentially

Multiple Fangs

Some vampire numbers have multiple pairs of fangs:

  • 126027 = 201 × 627 = 261 × 483

Prime Factorization

Vampire numbers are always composite (by definition) and often have interesting prime factorizations that relate to their digit properties.

Discovery and History

Vampire numbers were first described by Clifford Pickover in 1994, though the concept had been explored informally before. The name "vampire" comes from the idea that the number "bites" its fangs to create offspring with the same genetic material (digits).

  • Pseudovampire numbers: Similar but with relaxed constraints

  • Prime vampire numbers: Vampire numbers where both fangs are prime

  • Anagrams: The digit rearrangement property relates to anagrammatic numbers

Computational Aspects

Finding vampire numbers involves:

  1. Generating all possible factor pairs for numbers with even digit counts

  2. Checking if the digit multisets match

  3. Verifying the trailing zero constraint

The search becomes computationally intensive for larger numbers due to the exponential growth in possibilities.

Applications

While primarily recreational, vampire numbers demonstrate interesting properties of:

  • Digit manipulation algorithms

  • Factorization techniques

  • Combinatorial mathematics

  • Computer science algorithms for permutation checking