Some Prime Twin Philosophy
- I was exposed to this problem first in university. My professor said
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
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
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?