Nombres vampires

Vampire numbers have an unusually precise birth date for a piece of recreational number theory. Clifford A. Pickover introduced them in a 1994 post to the Usenet newsgroup sci.math, and wrote them up the following year in Keys to Infinity (1995), where chapter 30, "Vampire Numbers", occupies pages 227–231. The OEIS entry credits him by name. The conceit is that the number is a vampire and the two factors are its fangs: each hides half the victim's digits, and multiplying them brings the victim back.

Pickover was writing at the height of a brief vogue, in the early and mid-1990s, for naming recreational integer families after whatever image the pattern suggested — a style that reads as slightly embarrassing in a journal and works perfectly well in a popular book. The gimmick did its job: the definition is memorable, the smallest example is small enough to check in your head, and the search is just hard enough to be an interesting programming problem rather than a one-liner.

The one piece of the definition that looks arbitrary is the rule that the fangs may not both end in zero, and it earns its place. Without it, multiplying any vampire number by 100 produces another one for free — 1260 = 21 × 60 begets 126000 = 210 × 600 — and the four-digit list of seven would spawn seven junk six-digit entries. With the rule in place the counts are clean: 7 four-digit vampire numbers, 148 with six digits, 3,228 with eight. Pickover also defined looser variants, including pseudovampire numbers whose fangs need not be the same length, and the counts of the strict version are catalogued separately as OEIS A048935.

Propriétés principales

• The smallest vampire number is 1260 = 21 × 60.
• A vampire number must have an even number of digits, because its two fangs have the same length and together account for all of them.
• The fangs may not both end in zero; without that rule every vampire number would generate another by multiplication by 100.
• There are exactly 7 four-digit vampire numbers — 1260, 1395, 1435, 1530, 1827, 2187 and 6880 — then 148 with six digits and 3,228 with eight.
• Because digit sums are preserved modulo 9, a vampire number satisfies v ≡ x + y (mod 9) as well as v = x · y, which forces (x − 1)(y − 1) ≡ 1 (mod 9) and prunes roughly 93% of candidate fang pairs.
• Some vampire numbers have more than one pair of fangs: 125460 = 204 × 615 = 246 × 510 is the first, and 13078260 is the first with three.
• Every vampire number is composite by construction, since both fangs are at least 10.
• 2187 = 3⁷ = 27 × 81 is both a vampire number and a perfect power.

Où on les rencontre

• Pickover's Keys to Infinity (1995) devotes a chapter to them, and is the source most citations trace back to.
• Finding vampire numbers is a popular programming-exercise and code-golf task — Rosetta Code carries it as a standard problem with implementations in dozens of languages.
• They are a common teaching example for the difference between a naive search and a pruned one: the modulo-9 congruence turns an intractable scan into a feasible one.
• Prime vampire numbers, where both fangs are prime, are a studied sub-family; 117067 = 167 × 701 is an example, with both factors prime.
• The name belongs to a small tradition of deliberately lurid recreational-mathematics coinages from the 1990s, alongside Pickover's other inventions.