JSH: Prime gaps examples
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From: Dancing Ant on 30 Jul 2010 16:01 On Jul 21, 5:24 pm, JSH <jst...(a)gmail.com> wrote: > > No. So you have the prime residue axiom which says the primes HAVE NO > PREFERENCE. > > But logically you know they can't as composites are products of > primes. If say, primes modulo 3 had a preference for, say, 1, then > guess what? Composites would reflect the SAME PREFERENCE. They have > NO CHOICE. They are PRODUCTS OF THE PRIMES. > So, James, let's see if by using your "primes have no preference since otherwise composites would reflect the same preference" can be used to disprove a claim that "all the primes other than 3 are equal to 2 modulo 3". Look at all of the composites that are not divisible by 3. Half of them have an odd number of primes in their prime decomposition and half have an even number of primes in their decomposition. If all primes other than 3 were equal to 2 mod 3, then all of the composites with an odd number of primes in their prime decomposition would be equal to 2 mod 3, all of the composites with an even number of primes in their decomposition would be equal to 1 mod 3. So half would be 2 mod 3, half would be 1 mod 3. The preference for primes to be 2 mod 3 had *no* effect on the distribution across the modulo classes of the composites. *None*. In fact, you can characterize the composites that equal 1 mod 3 as those that have an even number of primes in their prime decomposition that are equal to 2 mod 3 (and any number of primes in their decomposition equal to 1 mod 3), while the composites that equal 2 mod 3 are those with an odd number of primes equal to 2 mod 3 in their decomposition (and any number of primes equal to 1 mod 3). And since there is no preference (to use your terminology) for having an even or an odd number of prime factors equal to 2 mod 3, then even if primes have a preference for one modulo class over the other (provided there is at least one prime equal to 2 mod 3) you would *always* see an even distribution for the composites across the modulo classes. Can you refute this? If you do not, then I can only conclude that you admit that your argument is simply wrong. -- Dancing Ant |