Choice of vampire with many fang pairs

June 20-24 2003 an efficient C program using Michael Scott's Miracl bigint library  made an exhaustive search and found 100025 different fang pairs for the 70-digit vampire number:


= 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 fang pairs are sorted by the smaller of the fangs.

The tested number was carefully chosen.
The number of divisors for a number is given by how many ways the prime factors can be combined. This is the product of the prime factor exponents with each exponent incremented by 1. A number with many divisors should have a lot of small prime factors.
The prime factorization of the vampire is:
28·310·5·72·112· 132·172·19·23·29·31·37· 41·43·47·53·61·67·71·73·79· 83·101·103·107·109·113·127·139· 149·163·167·173·179·181
The vampire has 9·11·2·34·228 = 4,305,167,843,328 divisors.

A number with more divisors than all smaller numbers is called highly composite.
The largest possible number of divisors for a 70-digit number is 10·6·5·4·34·227 = 13,045,963,161,600 divisors (3 times as many as the vampire).
This is for the 888th highly composite number at

The chosen vampire is likely to have several times more fang pairs than the highly composite number for the following reasons:
- The prime factor 5 only occurs once, meaning there is no "waste" on untrue vampire equations.
- The prime factor 3 occurs several more times than in the highly composite number. It follows from the modulo 9 congruence, that if the vampire is divisible by 81 then both fangs are divisible by 9. If 3 had multiplicity 5 then that would only leave 2 possible fang combinations for 3: the single "free" 3 in either one or the other fang.
- The vampire starts with the digit "1" (even "10"). The smaller 35-digit fang in a pair must be above vampire/1035, i.e. it must start with larger digits than the vampire. The first 9 fang pairs show slightly larger starting digits.
- Each digit occurs exactly 7 times, which should be the most common distribution in a factorization into two 35-digit numbers. This was accomplished with a computer search among "almost" highly composite 70-digit numbers satisfying the previous conditions. The search result was a vampire where the prime factorization is missing 7 primes below 181.