in our e-book Prime Passages to Paradise
by H. PeterAleff
Volume 1: Patterns of prime distribution
in "polygonal - number pyramids"
1.1 The misleading prime number sieve of Eratosthenes
Figure 1: Some square "Sieves of Eratosthenes" with primes and other special numbers marked
Figure 2: Rectangular arrays four and six units wide, and the distribution of primes into only two of their columns
1.2. Proving the twin prime conjecture
1.3. Laws and Order in Primeland
1.3.1. Numerical versus physical universes
1.3.2. The ongoing search for prime laws
1.4. Polygonal numbers
Figure 4: Polygonal numbers as dot figures, and some simple visual proofs of their relationships
1.4.1. Gnomons and triangle chemistry
1.4.2. Polygonal-number "pyramids"
1.4.3. The triangular- number "pyramid"
Figure 5: The top of the triangular- number stairway, and its alignments of primes
Figure 6: The same number- line segments as in Figure 5, but stacked symmetrically
1.4.4. The square- to- square "pyramid"
Figure 7: The apex of the square- to- square "pyramid", and its strings of primes
1.5. Prime-rich edge-parallels
Table 1: Angles of repetition for columns and edge parallels
1.6. More polygonal-number pyramids
Figure 8: Apex of pentagonal- number pyramid, with prime- solid diagonal strings
Figure 9: Apex of hexagonal- number pyramid with diagonals and prime- free columns
Figure 10: Apex of heptagonal- number pyramid, and its prime- solid diagonals
Figure 11: Apex of octagonal- number pyramid, with strings of twin prime centers
Figure 12: Apex of nonagonal- number pyramid
1.7. Structures of polygonal- number pyramids
1.8. Prime- rich columns in the first two polygonal- number pyramids
Figure 13: Patterns in the gaps between the primes of the triangular- number pyramid column at 7. Earlier column members return in a double cascade as factors in later ones.
Figure 14: Gaps between primes in the Euler Column under five, their composition from earlier primes, and their progression of "guest factors"
1.9. Prime- escorted triangular numbers as prime producers
Table 2: Curious connections between the triangular- number and square- to- square number pyramids
1.10. Last digits predict prime densities
1.11. Some prime- solid string lengths
1.12. A facet of Eulerís genius
1.13. The Euler Pillar centered in the square- to- square number pyramid
1.14. The prime calendar-clock for Sol III
(An abbreviated version of this chapter is posted here as "Prime calendar")
Figure 15: The upper clock- face of the "number calendar" centered in the square- to- square number- pyramid array
Figure 16: The lower clock- face of the "number calendar"
1.15. Numerical perfection for our planet
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