Quantum Gravity 218.6: The Excluded 3 Rule in Combinatorics Compared to Our PI Rule

From Osher Doctorow

Einstein's exclusion of higher derivatives than second order in his
Field Equation(s), and our exclusion of higher polynomial orders than
2 in y in dy/dt equations (yielding Riccati Differential Equation) and
the exclusion of higher polynomial orders in t than 2 in coefficients
of Generalized Exponential Function (GEF) suggests that "something is
wrong with 3 or more" across several disciplines with physical
applications.

Tanya Khovanova in arXiv: 0712.3509 math-CO 21 Dec 2007, "9 divides no
odd Fibonacci", does in fact that something is wrong with 3 or powers
of 3 in odd Fibonacci numbers which are of course obained from the
FIbonacci series or sequence which in turn is strongly related to
Probable Causation/influence because it corresponds to "memory" unlike
almost trivial "1 step Markov chain" memory and 0 memory of
Independent Probability-Statistics.

Khovanova poves that odd numbers n such that no Fibonacci number is
divisible by n are:

1) 9, 17, 19, 23, 27, 31, 45, 51, 53, ...

in increasing order, which contains 9 = 3^2, 27 = 3^3, etc.

This means that Fibonacci numbers F(n) or Fn exclude powers of 3^k for
k integer > 1. They do not exclude division by 3 itself, since for
example the Fibonacci numbers 3, 21, 144, 987 are divisible by 3, but
even they are rather few and far between so to speak. The first few
Fibonacci numbers are:

2) 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,
1597, 2584, 4181, 6765, 10,946, 17,711, 28,657, 46,368, 75,025,
121,393, ...

There are also some very strong relationships between Fibonacci
numbers and quadratics including in the generating function of
Fibonacci numbers. See Wikipedia's "Fibonacci number."

Osher Doctorow

.