CNB13 (PM2) Philosophical Musings – slogans that keep me going
Slogan 1: EVERYTHING CYCLES
Every (0,1)-pattern, found anywhere in T or C, will repeat, in several (perhaps an infinity of) different directions. And every 0, and every 1, in the pattern, has its repetitions satisfying, or governed by, its own Arithmetic Progression. But this set of APs has one thing in common: all the APs have the same common difference. There’s an invariance theorem for you!
The music of all these APs, seething and jostling along to eternity, resonating with their joint sounds and common periods of vibration, determines the symphony of the natural numbers. The music of the primes within this cacophony is a main theme, with its very own, basic charms. But, as slogan 2 below says, this theme is controlled, bounded by, and guided by, the patterns of the aforesaid APs.
The single or collective sounds of the primes is certainly not random noise; how can it be, if the humble Arithmetic Progression, with all its magnificent body of chums in every linked pattern, has some say as to when each single prime shall enter the fray, and introduce its own piping, thrumming sound to the orchestration?
Slogan 2: COPRIMENESS BEGETS PRIMENESS
Every 1 occurring in a (0,1)-pattern signifies a coprimeness between two natural numbers.
Therefore the (0,1)-pattern contains a lot of imformation about the coprimeness, and hence the primeness, of the numbers related to that pattern.
Slogan 3: POWERS OF PRIMES ARE IMPORTANT
When a prime-sieve (‘p-siv’ for short) is growing in its row of C, it arrives at the leading diagonal (l.d.) at precisely the same moment that a stalactite, its vertical ‘mate’, is descending (growing) in its vertical column and arrives there too. At that moment of time a prime cycle-number is born! Let us suppose this happens at the intersection of row Rj and column Cj. So the new prime is p(j) , and the pattern in Rj is 111…1110, its head 0 being on the l.d. This constitutes the fundamental cycle (f.c.) of the cycle-number p(j); the same pattern occurs vertically in Cj, by virtue of neck-tie cycling.
(to be continued)