Fibonacci Division

Now ready for the process of Fibonacci "division" -
- Since B and B' are the number of parts of each "level", let the total number of parts be called Y,
the next larger Fibonacci number after B and B' so that B + B' = Y.
- We may now say that -
- X divided into Y parts will become : B parts of A plus B' parts of A'.
The values X and Y may be any Fibonacci numbers whatever and the values A and A' (as well as the values B and B') will always be adjacent Fibonacci numbers.

From this it is possible to prepare a program which divides X into Y parts with two sizes of quotients (A and A') and two sizes of "sub-divisors" (B and B', which are not only adjacent Fibonacci numbers, but the two Fibonacci numbers below the main divisor Y).

EXAMPLE : 21 divided into 8 parts is 3 parts of 2 plus 5 parts of 3 (see illustration above)
EXAMPLE : 377 divided into 55 parts is 21 parts of 5 plus 34 parts of 8
EXAMPLE : 317,811 divided into 987 parts is 377 parts of 233 plus 610 parts of 377
Some dividends (values of X) will even "factor" as the sum of two squares -
EXAMPLE : 233 divided into 21 parts is 8 parts of 8 plus 13 parts of 13

 ... 13 -8 5 -3 2 -1 1 0 1 1 2 3 5 8 13 ...

This operation of division does not limit itself to positive numbers. The "mirror" section of the Fibonacci series can be produced by moving toward the left and by subtracting two adjacent numbers to produce the next number.
This will lead us right through the "central 0" and into the mirror section which, always producing similar values to those we had on the right, alternates positive and negative values.
Each alternate Fibonacci number (1, 2, 5, 13 ...) appears as a positive value in both sections. Whenever one of these numbers appears in our operation of "division" (either in the dividend or in the divisor) the possibilities of solution will be doubled.

EXAMPLE : 13 divided into 8 parts is
3 parts of 1 plus 5 parts of 2, also
3 parts of -144 plus 5 parts of 89
If both the dividend and the divisor have Fibonacci numbers which appear in both sections,
there will be four solutions to the "division".
EXAMPLE : 13 divided into 2 parts is
1 part of 5 plus 1 part of 8, also
5 parts of 3 plus -3 parts of -1, also
1 part of 34 plus 1 part of -21, also
5 parts of 89 plus -3 parts of 144
EXAMPLE : 4,181 divided into 4,181 parts is
1,597 parts of 1 plus 2,584 parts of 1, also
10,946 parts of 1 plus -6,765 parts of 1, also
1,597 parts of 24,157,817 plus 2584 parts of -14,930,352, also
10,946 parts of 63,245,986 plus -6,765 parts of 102,334,155
"Negative parts" will be examined more closely in a moment.
Negative dividends may also be divided into any number of parts.
EXAMPLE : -55 divided into 8 parts is 3 parts of 610 plus 5 parts of -377

Positive or negative Fibonacci numbers may be divided into any negative Fibonacci number of parts.
One could use the analogy that quotient values (positive or negative) can be placed in convex piles (positive parts) or in concave piles (negative parts).
Negative values in a concave pile would evidently produce a positive value.

EXAMPLE : 377 divided into -3 parts is -8 parts of 4181 plus 5 parts of 6765
EXAMPLE : -987 divided into -1 parts is -3 parts of 233 plus 2 parts of -144 (see illustration above)

These divisions seem to operate just as well with larger numbers and one is led to believe that the size of integer which a program or a computer can handle is the only limit to the size of values within this program.
EXAMPLE : 63,245,986 divided into -267,914,296 parts is
-701,408,733 parts of 3 plus 433,494,437 parts of 5, also
-701,408,733 parts of 61,305,790,721,611,591 plus 433,494,437 parts of 99,194,853,094,755,497

1 Divided By 1
Since the Fibonacci number 1 appears three times in the complete series, there will be three solutions every time it is used (either as dividend or as divisor) and nine solutions if it is divided by itself.
1 divided into 1 part can therefore be :

 plus 1 times 1 0 times 0 plus 1 times 1 0 times -1 plus 1 times 1 1 times 1 plus 0 times 2 1 times 1 plus 0 times 1 1 times 1 plus 0 times 0 2 times 3 plus -1 times 5 2 times 2 plus -1 times 3 2 times 1 plus -1 times 1

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