Format of tables

We follow the format in [BMS-1988].

First note that F2n=FnLn. Thus it suffices to only list Fn for n odd.

The factors of a Fibonacci number can be classified as algebraic or primitive. Only the primitive factors need be listed in the tables, as the algebraic factors are the primitive factors of smaller Fibonacci numbers so only their subscripts need be listed. We place these subscripts in a parenthesized list.

For example, the factorization of F105 can be written as

F105  (3,5,7,15,21,35) 8288823481
Here (3,5,7,15,21,35) are the subscripts of the Fibonacci numbers whose primitive parts are the algebraic factors of F105 and 8288823481 is the only primitive factor of F105.

Since the factorizations of F3, F5, F7, F15, F21, F35 can be listed as

F3  2
F5  5
F7  13
F15 (3,5) 61
F21 (3,7) 421
F35 (5,7) 141961
the factorization of F105 is obtained by collecting the primitive factors from these lines and combining it with the primitive factor of F105 to give
F105=2.5.13.61.421.141961.8288823481

One additional compression of the tables is done. The largest primitive factor is not listed but rather indicated by "Pxxx" or "Cxxx" depending on whether the factor is prime or composite and "xxx" is the number of decimal digits. This factor can be reconstructed by calculating Fn and dividing out all the other factors. Thus the actual entry for F105 is

F105  (3,5,7,15,21,35) P10
The table of Lucas factorizations follows the same format with one difference. Because of a special identity discovered by Aurifeuille, there exists the Aurifeuillian factorization of Lucas numbers

L5n=LnA5nB5n, n odd

where

A5n=5Fn2-5Fn+1

and

B5n=5Fn2+5Fn+1

Thus, for example, L105 is listed as

L105  (3,7,21) A.B
L105A  (15,35B) P5
L105B  (5,35A) P6

Intrinsic factors

Some of the factors that we have listed as primitive really are algebraic and are called intrinsic factors. (For the formal definitions of algebraic and primitive see [BMS-1988].)

If p is a primitive factor of Fm (resp. Lm), then p will be a factor (an intrinsic factor) of Fn (resp. Ln) when n = prm where r >= 1.

Intrinsic factors are denoted in the tables by a "*" following the factor. For example,

F125 (5,25) 5*.P21

Other theorems

There are also theorems related to the factorization of Fibonacci and Lucas numbers that are not covered by the above "standard format".


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