
The Largest Versatile Number
Posted:
Jun 22, 1996 8:03 PM


The Largest Versatile Number
by William Lauritzen
Versatile Numbers are numbers that have more factors than any smaller number. Twelve is a versatile number because 12 has 6 factors and no number less than 12 has that many factors. These are the first few versatile numbers: 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 840, 1260, 1680, 2520, 5040, ... In this brief paper I discuss efforts to find the largest known versatile number.
In the paper "The Versatility of Numbers" (Dome, Spring 1996) I much more fully discuss these numbers. This paper is available from me or the magazine "Dome." However, I will give a short synopsis here of the history of versatile numbers as I know it.
Although I independently discovered these numbers, I found that the famous Indian mathematician Ramanujan had investigated them and had called them "highly composite" numbers. This name is not very communicative and as a result I believe very few people know about these numbers. The English mathematician Hardy said these numbers were "as unlike a prime as a number can be." I like this statement because I sometimes refer to these numbers as antiprimes (although I prefer the name versatiles).
In examining these numbers one can see that certain of them (such as 12, 24, 36, 60, and 360 are used by modern civilization to divide up time, distance, goods (the dozen), and the heavens (the circle). In other words some ancient civilization (whether it was the Summarians or the Egyptians or another civilization is unimportant for our purpose here) picked versatile numbers to use. Perhaps through trial and error someone discovered the advantages of these numbers, and perhaps they even defined this class of numbers. No one really knows for sure how the first use of them started.
Meyer Rainer, a scholar from Switzerland, recently communicated to me that Plato considered the versatile number 5040 as "an ideal number of citizens in an ideal community, where everyone lives in peace, freedom, and friendship, and all measurements, weighings and partitions are done in the proper way." Although I don't necessarily totally agree with Plato, this is interesting nonetheless and I am investigating this further as this parallels my own reseach.
In "The Versatility of Numbers" I point out that these numbers CAN BE SHARED EVENLY EASIER than any other numbers, and propose that IGNORANCE OF THIS CLASS OF NUMBERS MAY HAVE INCREASED THE AMOUNT OF VIOLENCE IN THE WORLD. If so, then VERSATILE NUMBERS ARE AT LEAST AS IMPORTANT AS PRIME NUMBERS and should be treated so.
Imagine some children: The have some apples to share between them. If they have twelve apples, a versatile number, the apples could be shared equally with 2, 3, 4, or 6 children. No fighting or bloody noses. If they have 10 apples, a nonversatile number, the apples can only be shared equally if there are 2 or 5 children. Unless they know how to make fractions of items easily and quickly, I believe fighting and bloody noses would be significantly more likely. This could be tested by social psychologists. A special observer, trained to note incidences of aggression, yet who knows nothing about the purpose of the experiment, would be used.
I don't believe that the awareness and use of versatile numbers will eliminate all of humankind's aggressive actions. Rather I believe that their use will LUBRICATE SOCIAL INTERACTION. So my goal is to stimulate experimenters to verify or disprove this proposition, and if it is verified, to have these numbers taught in all schools right alongside prime numbers. In the meantime, do your own experimenting, to discover the uses of versatile numbers.
The largest versatile number has 13 digits. The largest prime has 258,716 digits. That great disparity in size between the largest prime and the largest versatile may symbolize to some degree the lack of attention society has given versatiles compared to primes. (However, one can find a large prime without knowing all the previous primes and one cannot find a large versatile without knowing all the previous versatiles.)
The previous slargest versatile number was published by Ramanujan in 1915. The number was 6,746,328,388,800 and it has 10,080 factors. I may be wrong but I believe no one else has found any larger ones. I got interested in verifying his work and trying to find larger ones. Building a car that can break the landspeed record can give insight into automotive design, and by the same logic, finding the largest versatile can give insight into these numbers.
Of course Ramanujan didn't have the advantages of computers. However, it can be quite frustrating trying to write a program that will generate versatiles as these numbers, like their opposites the primes, can not be predicted by any rule.
In any case, on June 21, the longest day of the year, with help from Dan Hurt, I found two additional numbers.
LIST OF 102 KNOWN VERSATILE NUMBERS
Versatile # # of Factors 2 2 4 3 6 4 12 6 24 8 36 9 48 10 60 12 120 16 180 18 240 20 360 24 720 30 840 32 1260 36 1680 40 2520 48 5040 60 7560 64 10080 72 15120 80 20160 84 25200 90 27720 96 45360 100 50400 108 55440 120 83160 128 110880 144 166320 160 221760 168 277200 180 332640 192 498960 200 554400 216 665280 224 720720 240 1081080 256 1441440 288 2162160 320 2882880 336 3603600 360 4324320 384 6486480 400 7207200 432 8648640 448 10810800 480 14414400 504 17297280 512 21621600 576 32432400 600 36756720 640 43243200 672 61261200 720 73513440 768 110270160 800 122522400 864 147026880 896 183783600 960 245044800 1008 294053760 1024 367567200 1152 551350800 1200 698377680 1280 735134400 1344 1102701600 1440 1396755360 1536 2095133040 1600 2205403200 1680 2327925600 1728 2793510720 1792 3491888400 1920 4655851200 2016 5587021440 2048 6983776800 2304 10475665200 2400 13967553600 2688 20951330400 2880 27935107200 3072 41902660800 3360 48886437600 3456 64250746560 3584 73329656400 3600 80313433200 3840 97772875200 4032 128501493120 4096 146659312800 4320 160626866400 4608 240940299600 4800 321253732800 5376 481880599200 5760 642507465600 6144 963761198400 6720 1124388064800 6912 1606268664000 7168 1686582097200 7200 1927522396800 7680 2248776129600 8064 3212537328000 8192 3373164194400 8640 4497552259200 9216 6746328388800 10080 (all prior to here published in 1915)  (all after here found in 1996) 8995104518400 10368 9316358251200 10752 Thanks to Jim Barbour, Mike Easterbrook, Dan Hurt, Michael Kenniston.
William Gunther Lauritzen 809D East Garfield Glendale, CA 91205

