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Old 2006-11-01, 11:59   #79
troels munkner
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May 2006

29 Posts

Originally Posted by mfgoode View Post

I reiterate Troels definitions.

2) The Never Primes: These comprise all even numbers AND all odd numbers divisible by 3
On the number line NP are located symmetrically around 0 and so may be called 0-centrred integers.
NP constitute 2/3 of all numbers including two real primes No.s 2 and 3.

3) Possible Primes (PP): These are all odd numbers which cannot be divided by 3.
PP are located symmetrically around +1 or – 1 depending on your choice.
These may be called 1-centred integers.
Dear Malcolm,
Thanks for your replies to other mathematicians and to me.
I have used +1 as the centre for all primes and prime products, and it
has a number of advantages.
The expression ((6*M)+1) comprises all primes and prime products,
M being any or all of the natural numbers from - infinity to + infinity.
((6*(-39)+1) = -233, which´is a prime
((6*(+39)+1) = 235, which is a prime product.

A prime product such as ((6*M)+1) * ((6*N)+1) = 36 (NM) + 6*(M+N) + 1
is an integer, which will never be divisible by 2 or 3. Conclusion: 2 and 3
could be called anything but "primes".

N (just as M) being any or all natural numbers from - infinity to + infinity.
(+) * (+) is of course (+), (-) * (-) will also give a (+) integer,
(+) * (-) will give a (-) integer.
All primes and prime products are divisible by 1 (i.e. N=0).
If you want to look for (M) and (N), which means to factorize a possible
prime, you can do it by subtracting 1 from the integer in question and then
use a second order equation to find or not find the two roots (M) and (N).
If the sum of (M+N) is odd and > 1, the factorization results in
(Even integer)^2 - 3^2 * (an odd integer)^2.
If the sum of (M+N) is 0 or any other even number, the factorization ends in
(Odd integer)^2 - 6^2 * (any integer, including 0)^2.

The sign of a prime or prime product can easily be predicted by modulation
(modulo 9), and it is easy to show if the sum (M+N) is even or odd.

If you have the time you can try to follow my ideas:
7*13 = 91 = 10^2-3^2 etc.
7*19 = 133 = 13^2-6^2 etc.

I am not drowning. I will in fact consider to reflect to the many harsh replies,
which I have received (directly or indirerctly), but maybe it is not worth
the effort. A famous citation from Schiller's Jeanne d'Arc comes to my mind

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