n         F(n)

 125   59425114757512643212875125 

 184   127127879743834334146972278486287885163 

 

 from

 

    4   kill "pandfibo.txt":open "pandfibo.txt" for output as #2

    5   A=1:B=1:Lvl=2

   10   while C<9999999999999999999999999999999999999999

   20     C=A+B:A=B:B=C:inc Lvl

   30     S$=str(C)

   50     Good=1

   60     for I=1 to 9

   70        if instr(S$,cutspc(str(I)))=0 then Good=0:cancel for:goto 90

   80     next

   90     if Good then

   91       :if instr(S$,"0")=0 then print C:Ct=Ct+1:print #2,Lvl,C

  100   wend

  110   print Ct:close #2

The 3-digit numbers preceding the F numbers are the ordinal position of these Fibonacci numbers, in the system where F(1)=F(2)=1.

No further results come from even over 200 9's in line 10. The probability of avoiding any zeros is probably too low with increased numbers of digits. In fact even if C is allowed to go up to 10^999, no further examples are found. There could be a hypothesis that there are no further examples beyond these two (proof anyone), or a probability that no further examples exist, based on an infinite series of probabilities for the infinite series of Fibonacci's that increase in length at a regular rate.