100/89= 1.1235..., which includes the first five Fibonacci numbers.

10000/9899=1.010203050813213455... includes the first ten Fibonacci numbers.

1000000/998999=1.001002003005... produces the first 15 Fibonacci numbers.

If you add two zeros in the numerator and two nines (one at the beginning, one at the end) in the denominator, does this Fibonacci production go on?

The following program verifies that at the next step, indeed the 1- through 4-digit Fibonaccis continue, until 6765 is converted to 6766 by a carry from the next, 5-digit, Fibonacci:

list
   10   print 100000000/99989999
   20   A=1:B=1
   30   repeat
   40     C=A+B:A=B:B=C:print C
   50   until C>9999
OK
run
 1.00010002000300050008001300210034005500890144023303770610098715972584418167660
9477713866163755036
 2
 3
 5
 8
 13
 21
 34
 55
 89
 144
 233
 377
 610
 987
 1597
 2584
 4181
 6765
 10946
OK

But I'm sure we all want to know why.

Posted by Charlie on 2007-02-22 16:02:20