# Format of tables

We follow the format in
[BMS-1988].
First note that *F*_{2n}=F_{n}L_{n}. Thus
it suffices to only list *F*_{n} 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 *F*_{105} 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 *F*_{105} and
8288823481 is the only primitive factor of *F*_{105}.
Since the factorizations of *F*_{3}, *F*_{5},
*F*_{7},
*F*_{15}, *F*_{21}, *F*_{35}
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 *F*_{105} is obtained by collecting
the
primitive factors from these lines and combining it with the primitive
factor of *F*_{105} to give
*F*_{105}=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
*F*_{n} and dividing out all the other factors.
Thus the
actual entry for *F*_{105} 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
*L*_{5n}=L_{n}A_{5n}B_{5n},
*n* odd

where

*A*_{5n}=5F_{n}^{2}-5F_{n}+1

and

*B*_{5n}=5F_{n}^{2}+5F_{n}+1

Thus, for example, *L*_{105} 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 *F*_{m} (resp. *L*_{m}),
then *p* will be a factor (an *intrinsic* factor) of *F*_{n}
(resp. *L*_{n}) when *n = p*^{r}m 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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