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From: OsherD on 30 May 2010 21:57
From Osher Doctorow

With the disclaimer that the Austrian-Hungarian paper to be cited did
not explicitly state "Probable Causation/Influence (PI)" as names, the
researchers Akos Pinter of U. Debrecen Hungary and Volker Ziegler of
Graz U. of Technology Graz Austria have in fact related the Fibonacci
sequence to PI form via 3-term arithmetic progressions (3TAP for short
here) in binary recurrences. Their paper is:

1) "On arithmetic progressions in recurrence - a new characterization
of the Fibonacci sequence," Akos Pinter and Volker Ziegler," in arXiv
May 2010 (I will supply the exact citation shortly).

Recall the Probable Causation/Influence (PI) has the form:

2) 1 + y - x, 0 < = y < = x < = 1

With normalization, the theorems and corollaries of Pinter and Ziegler
involve expressions of type:

3) X^a - 2X^b + 1, X^2 - X - 1 ( = -(X - X^2 + 1)), X^d -
a_(d-1)X^(d-1) - ... - a_o (Companion Polynomial), etc.

Notice that in their Corollary 1, the expression:

4) X^a - 2X^b + 1, a > b > 0

occurs. When X is in [0, 1] as with Probable Causation/Influence,
then X^a < = Xb for a > b > 0, as for example (1/2) > (1/2)^2 = 1/4.
So (4) has indeed the correct form 1 + y - x for 0 < = y < = x < = 1
(if normalized into [0, 1]).

Recent applications of arithmetic progressions cited by the above
authors include points on elliptic curves, solutions of Pellian
Equations (Pell's Equation and its generalizations, which are also
related to PI from past posts in this thread), solutions of norm form
equations.

Expressions of type (3) are not the only ones in their paper, but
others tend to be either slight modifications of type (3) or more
complicated expressions that I will try to discuss later.

Osher Doctorow
From: OsherD on 30 May 2010 22:35
From Osher Doctorow

The detailed numbers of the paper by Pinter and Ziegler are:

1) arXiv: 1005.3624 v1 [math.NT] 20 May 2010, 20 pages.

The authors show that the Fibonacci sequence is the unique binary
recurrence containing infinitely many 3TAPs, where a linear recurrence
fn of order d is:

2) f_(n+d) = a_(d-1)f_(n+d-1) + ... + a_o f_n with a_i complex
numbers for i = 0 to d-1, a_0 not 0, and the sequence doesn't satisfy
such an equation with fewer terms, the f_n being complex, n = 1, 2,
3, ....

The Companion Polynomial (CP here) of f_(n+d) plays an important role,
and is defined by:

3) P(X) = X^d - a_(d-1)X^(d-1) - ... - a_o

For various theorems and corollaries, these turn out to include:

4) X^a - 2X^b + 1 (as a factor), X^2 + X - 1, X^2 - X - 1, X^2 + 2X +
2, etc.

Osher Doctorow
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