that he needed to see a base case and a method from getting from k to

k+1 (induction). What I couldn't express to him at the time was that

no such function existed because the creation of primes was and is

the only true random function that I've ever been exposed to. To

have a method to go from k to k+1 breaks that assumption of

randomness.

The numbers 2 and 5 are very nice numbers because they predictably

destroy, in the sieve, all even numbers and all 5 ending numbers.

Nice of them. But that's the end of the predictability.

3 is a crucial number in that it defines the first infinite pattern

of possibilities for primes. The pattern of opportunities for prime

twins that the number 3 creates is well known. In the sieve of odd

numbers it looks like

100100100100100100..., were 1 is a multiple of 3 and 0 isn't (hence

an opportunity for a prime)

Standard combinatorics yields that for any prime p, the there will be

opportunities for prime twins in the resulting pattern (proof is

trivial, but can be provided).

Ok, let's have some fun now: Assuming from the above that for any

prime number p, the pattern (sieve) has opportunities for prime

twins... prime twins will occur whenever the prime twin opportunity

lies between p and p^2. If only we could prove that such an

opportunity will always eventually lie between some n and n^2 where n

is prime. We can't. But don't get dejected.

We can't because of something magical... the pattern produced by

infinitely adding new primes to the sieves is RANDOM. Not, pseudo

random, but really random. Up until recently I only believed

in 'patterns sufficiently complex to avoid simple categorization'.

This process is really random!

The pattern produced by 3 is simple: 100 (size=3). The pattern

produced by 3,5 is more complicated 110100100101100 (size=3*5=15) but

still predictable. 3,5,7 is more complicated still. As we continue

to add primes to the pattern (producing longer, more complicated

patterns) at any given step, the resulting pattern is more

complicated than the previous p-1 pattern were p-1 is the previous

prime. As we continue, and add primes to the pattern without bound,

we produce the a random pattern, one with infinite complexity and

zero predictability.

Assume you swallow the previous argument (that the patterns of primes

and prime twins et all are random, btw it took me 5 years to swallow

this argument) if the patterns of prime twin opportunities are random

then so too is the distance of the prime twin opportunity from it's

largest prime contributor p. So take a pattern pat1 where the first

prime opportunity is beyond c^2 and c is the largest prime

contributor to that pattern. Well we won't get a prime twin then,

but forunately the algorithm continues. We have n->inf iterations of

this algorithm until the pattern becomes random.

In a random process you can't guarantee me that the first prime

opportunity will NEVER come between c and c^2 and therefore it will

eventually come there. Unless you can describe p such that after

that prime p, no prime twin opportunities exist. Since there is no

such p that removes all prime twin opportunities, and the process of

producing new sieves in iterations without bounds produces random

patterns, there will always be a prime twin opportunity which

eventually lies between c and c^2.

The same randomness which caused our difficulty in proving this

problem now then solves it. How do you like that?