reply to post by stander
That's strange that you couldn't grasp the meaning of "near infinity," when you use the term "infinity of primes," which is very misleading, coz
infinity is not a number, nor is it a synonym to "bunch of" or "plenty of" primes.
Nothing strange about it as in maths, the term "infinity", usually denotes an unbounded limit.
In this thread, we are concerned only with the set of positive integers n, where n grows without bound, i.e. Ip = {1,2,3,4,5, ...., n}.
By saying "near infinity", one can then reasonably ask "just how close is that really to infinity ... are we talking withing spitting distance ?",
as if the term "infinity" itself represented some kind of super-duper humongous number.
But no matter how close you may like to think you've chosen a number close to "near infinity", there will always be an infinity more of numbers
between the number you picked and .... well, infinty, I guess :-)
Don't make the common mistake of thinking that the concept of "infinity" can be equivalent to a number, because it can't.
That's like asking what happens when I divide 10 by infinity, or multiply 5 by infinity ... can't do it because infinity is NOT a number ...
therefore NOT being a number, you can't get "near to it".
Anyway, let's not worry too much about semantics ... ok ?
Let me see if I can answer your question:
Here we'll use modular arithmetic in base 6.
If we assume that 2 and 3 are primes, then it naturally follows that all multiples of 2 and 3 greater than 2 and 3 are not primes.
In mod 6, there are six subdivisions of the positive integers (n)
0 mod 6 = 6n and giving answers such as 6, 12, 18, 24, 30, ...
1 mod 6 = 6n + 1 and giving answers such as 1, 7, 13, 19, 25, ...
2 mod 6 = 6n + 2 and giving answers such as : 2, 8, 14, 20, 26, ...
3 mod 6 = 6n + 3 and giving answers such as : 3, 9, 15, 21, 27, ...
4 mod 6 = 6n + 4 and giving answers such as :4, 10, 16, 22, 28,...
5 mod 6 = 6n + 5 and giving answers such as : 5, 11, 17, 23, 29, ... (Note that 6n+5 is equivalent to 6n-1)
Now, any positive integer that is equivalent to 0 mod 6, 2 mod 6, or 4 mod 6 must be even and therefore will be divisible by 2.
e.g 6n/2 = 3n, (6n + 2) / 2 = 3n + 1, (6n + 4) / 2 = 3n + 2.
This means that any such positive integer can't be prime unless it's actually the number 2.
Also, any positive integer that is equivalent to 3 mod 6 is divisible by 3.
e.g. (6n + 3) / 3 = 2n + 1.
This means that any such positive integer can't be prime unless it's actually the number 3.
So after having eliminated the above, this leaves us with the conclusion that any positive integer that is equivalent to 1 mod 6 (i.e. 6n+1) or 5 mod
6 (i.e. 6n-1) COULD POSSIBLY be prime.
This is because not all numbers of the form 6n + 1 or 6n + 5 (i.e. 6n-1) are prime, but definitely every prime that is greater than three will be
equivalent to one of those two forms.
Hope I've been reasonably clear in the above steps