Steve Whealton








Banding with Prime Index Numbers

n

  P(n)

The prime numbers get scarcer as you climb through the integers. Mathematicians have studied this fact for many centuries.

The first thing that we will make use of is the notion of the index number of some given prime.

For several valid reasons, both one and zero have been defined to be neither prime numbers nor composite numbers. We reflect this by giving both of them a rating of zero. So P(0) = 0 and P(1) = 0.

2 is the smallest prime. So its primacy will be rewarded by assigning to it the prime index number of 1. P(2) = 1.

3 is also a prime,so it should get the next available number, which is 2. This reflects the fact that 3 is the 2nd prime. P(3) = 2.

4 is not a prime, so it does not deserve a new prime index number of its own. It will have to make do with 2.

Somewhat more precisely, 2 can be said to be the index number of the largest prime (3) equal to or smaller than 4. It is this basic idea that will be used in determining the prime index numbers for all larger integers.

5 is the next prime, so it earns, and receives, index number 3.

6 inherits 3, just as 4 inherited 2.

7 gets 4, fresh; 8, 9, and 10 inherit 4. 11 gets 5 anew, and 12 gets it by inheritance, so to speak.

This process can continue indefinitely. The image, prime banded 1, makes use of this kind of banding. "prime banded 1" was created by beginning with a crisper and more detailed image. The prime banding method of shrinking the numbers in that detailed image eliminated much of the detail in that original image.

For me, "prime banded 1" looks as though a more detailed image had originally been present, perhaps made out of wax. It is as if the prime banding procedure has melted away much of the wax, leaving only an outline of the original.

Other criteria that, like the prime criterion, include some numbers and exclude others, can also be used. The so-called "lucky numbers" were defined so as to be distributed much as the prime numbers are. So they suggest a good alternative system for banding

Still other criteria might include and exclude numbers according to quite different overall patterns. In inventing or in choosing such criteria, care should be taken not to make things too orderly. It might be interesting also to search for criteria that will produce distributions that are quite different from the primes' and the luckies' distributions.

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