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The Largest Versatile Number
Posted:
Jun 22, 1996 8:03 PM
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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
none-the-less 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 non-versatile 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
809-D East Garfield
Glendale, CA 91205
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