Perhaps the finer details of the results are interesting, but surely
the main result of prime number generation is self evident?
Essentially what is being said is this. Define a set X of numbers
x0, x1, x2, x3, xm, ....
2* xm +1 is always composite and includes all composites.
If we take any y which is not in the set X, then by definition
2*y + 1 cannot be composite, namely it must be prime.
Yes, Mark, the main result is self evident.
But in my opinion, it has some interest. It's somewhat like saying "erase all the
pentagonal numbers and the remaining numbers n all generate the prime numbers with 2*n+1".
Well, these numbers have two parameters ( m,n ) and therefore they are much less rare than
pentagonals (in fact, it's easy to know their density using the PNT), but you see my
point: a "simple" formula is involved.
It's a different way to get the same result from always, and I think we can totally
automatize them, get the pairs totally ordered with a simple algorithm or a simple set of
rules and then get some conclusions about them. For example:
If we could prove that for a fixed n0 we can always find a pair (m,n), such that the next
pair (m',n') bigger than it has D(m',n') >= 2+D(m,n), then we would have proven the twin
In general, note that if (m',n') is the next pair for (m,n) in this order, then we know
that the odd numbers between 2*D(m,n)-1 and 2*D(m',n')-1 are all prime. The interesting
thing is this order, in my opinion.
Of course, I can be totally mistaken and this order could be impossible of determining
with simple conditions, and to work on it worth nothing; or it could be obvious but
without real practical uses or untrivial information. While I arrive to one of these
conclusions, I'll keep thinking about this subject in my free time (well, among other
subjects as well :P).
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