I have lost the original post. Who was looking for a project
in computer science? In any case, shall I just post the idea
here? Contact me offlist for questions if you wish.
Prime numbers are very much a phenomenon of nature, or
rather meta-nature for the primes are the same for every
intelligent species that has discovered them, regardless
of what noxious fumes they respire or what number base
they may use. Primes are universal.
This idea is related to a function I have been toying with
since the publication of this month's Scientific American.
The Mathematical Recreations column describes the
gaps between primes, then defines the most common
interval as the "jumping champion".
The first several primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23,
29, 31, 37, 41... so the intervals in those first twelve primes
would be 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4. In that interval, the
most common jump is 2, so the jumping champion for the
first 13 primes is 2, so J(13) = 2.
Nowthen, this function as defined in the SA article is
rather dull. You go out very far and the J(n) = 6 for a
looooong long time, until you get way out into the jillions.
But if you define the J function with two parameters, a
lower and an upper limit, interesting things happen. For
instance if we wished to calc J(7, 13), the jumping champion
between the 7th and the thirteenth primes, then the primes
are 17, 19, 23, 29, 31, 37, 41, the jumps are 2, 4, 6, 2, 6, 4
and the jumping champion is a threeway tie between 2,
4 and 6. For greater interest, let all ties go to the
largest, so J(7, 13) = 6.
Nowthen, heres the trick. Make a plot, the x axis is lower
limit, the Y axis is the upper limit minus the lower limit. At
each X,Y coordinate is the jumping champion in that
interval J(X,Y). So on our map, at the coordinates (7,[13-7])
is a column 6 high.
Nowthen, if you write an algorithm to create a map of
all the jumping champions in all intervals, you create
a universal landscape of sorts, one that would be recognized
by mathematicians from the planet Ork as well as here.
It is a fascinating primescape. As I said before, the most
common jumping champion is 6, so let 6 be identified
as "sea level" on the primescape. You form a coastline,
where the upper limit minus the lower limit = 1. Along
this coastline are towers and mountains of various heights.
It really is a marvelous sight. There are islands, isolated
"land masses" in a sea of 6. There are lakes, which are
closed areas of 6 within areas of raised areas. The
primescape contains lagoons, inlets, rivers, mesas,
islands, everything. Again, that map that you create
is truly *universal* in that it is the same map as can
be created anywhere in the universe.
Computer science student, you project will be to
write a piece of code that you can distribute over
email to volunteers, who will pool their computer
resources to map out the primescape by each taking
a range of X and Y to map. Then you
will need some means of storing all the findings.
I have some ideas on how this could be done,
and I have already written the code to create the
map. With enough volunteers, perhaps 30, you
should be able to create a map between 1 and
about 1 million, and about 200 pixels out to sea.
The extropian list alone should get ya 10 or 20
volunteers. Then answer the following questions
about the primescape:
1) what is the largest island below X=1E6?
2) what is the island farthest from the coast?
3) what is the largest landlocked lake?
4) if volume is defined as area times altitude,
what is the landmass with the greatest volume
below X=1E6?
5) where is the steepest cliff?
6) where is the landmass with the greatest
number of different altitudes?
7) Are there any examples of an island within
a lake? [answer = yes] How many are there
of these below X = 1E6.
All of these answers may be known to some
extraterrestrial beast, but no human knows them.
It would make a wicked cool computer science
project. spike
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