Sunday, August 06, 2006

 

JSH: Primes and no preference

I have created enough threads that I am bothered by it but I need to emphasize some things that are important as this is a crucial juncture.

First off, why do the primes not show a preference for a particular residue modulo some other prime?

Like, as every prime other than 3 has a residue of 1 or 2 modulo 3, as, for instance, 5 has a residue of 2 modulo 3 since 5-3 = 2, why is there no possibility of preference for a residue of 2 modulo 3 over the residue of 1 modulo 3 which 7 has as 7-6 = 1?

Well, you look at the integers in general and notice that they advance by 1.

So, you have 5, 6, 7, giving you each of the possible residues modulo 3, and then you repeat—8, 9, 10—same pattern..

That goes out to infinity.

It gives no room for preference, as either you have primes or the product of primes, and if the primes could decide something—behave with human stubborness—and try to pick a residue, that pattern out to infinity could not be possible, as, for instance, if primes liked 1 modulo 3, then composites would tend to be 1 modulo 3 versus 2 modulo 3.

Unfortunately mathematicians like to play silly games like talk about "pure" ideas versus practical ones, and probability is considered practical in various areas of human endeavor, including gambling, as well as the sciences, and it seems to me they thought it beneath them to consider probabilistic approaches relying on this intrinsic reality of primes.

And it can seem a bit odd, like, how can there be a probability that 100 is prime? It's not.

But one can still use this idea over an interval to see how many primes one would expect, as I've demonstrated in posts showing the applicability of this idea, or you might consider some far off interval, or some rather large number and consider that probability.

Yes, it is definite whether or not a number is prime. But that definiteness goes out to infinity.

There are an infinity of primes, and each one of them is definitely a prime, and you will never know most of them.

Importantly, understanding this intrinsic reality of how primes relate to each other—with complete indifference—in terms of residues, allows you to construct logical arguments considering certain problems considered open, like the twin primes conjecture, and Goldbach's conjecture.

Understanding that to the primes it is of no consequence how they line up with regard to residues modulo other primes it is trivial to show that the twin primes conjecture must be true, and Goldbach's conjecture must be false—though counter-examples to his conjecture are unlikely to ever be found as the probability is so low.

That is, they exist, but how do you know where to look? Somewhere out there in infinity there are an infinite number of composites for which Goldbach's simple idea fails, but how do you pick one, and even if you did, is checking it within the range of human ability?

Since the results follow from primes not caring about how they relate to each other by residue, there is no other approach available and no other means of proof one way or the other.

All such attempts, logically, are doomed to fail, as the mathematical reason is just this necessity of prime independence.

Counter arguments are welcome. A proof is not refutable. There is no threat from objective responses questioning the logical chain presented.

I say, the logic is absolute. If that's wrong, prove it. Show a break in the logical chain.





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