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 wwwhomes.uni-bielefeld.de/achim/highly.txt
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.