When the digit is placed at the beginning and end of a positive integer N, the new number is 99*N.
Find the smallest value of N.
(In reply to
More precision with UBASIC computer program by Charlie)
Noticing the first 5 digits of N being the first 5 Fibonacci numbers, one sees that the next one is 1 higher than the next Fibonacci number only because it is receiving a "carry" from the next digit over.
The below table shows the result of keeping track of a number, each successive value of which is the preceding value multiplied by 10 and to which the next Fibonacci number has been added after the multiplication by 10. Up to the last few digits they track the digits of N, either for the 22-digit case with appended digit 1 or the 66-digit case with appended digit 1:
1 11
2 112
3 1123
5 11235
8 112358
13 1123593
21 11235951
34 112359544
55 1123595495
89 11235955039
144 112359550534
233 1123595505573
377 11235955056107
610 112359550561680
987 1123595505617787
1597 11235955056179467
2584 112359550561797254
4181 1123595505617976721
6765 11235955056179773975
10946 112359550561797750696
17711 1123595505617977524671
28657 11235955056179775275367
46368 112359550561797752800038
75025 1123595505617977528075405
121393 11235955056179775280875443
196418 112359550561797752808950848
317811 1123595505617977528089826291
514229 11235955056179775280898777139
832040 112359550561797752808988603430
1346269 1123595505617977528089887380569
2178309 11235955056179775280898875983999
3524578 112359550561797752808988763364568
5702887 1123595505617977528089887639348567
9227465 11235955056179775280898876402713135
14930352 112359550561797752808988764042061702
24157817 1123595505617977528089887640444774837
39088169 11235955056179775280898876404486836539
63245986 112359550561797752808988764044931611376
102334155 1123595505617977528089887640449418447915
165580141 11235955056179775280898876404494350059291
267914296 112359550561797752808988764044943768507206
433494437 1123595505617977528089887640449438118566497
701408733 11235955056179775280898876404494381887073703
1134903170 112359550561797752808988764044943820005640200
1836311903 1123595505617977528089887640449438201892713903
2971215073 11235955056179775280898876404494382021898354103
4807526976 112359550561797752808988764044943820223791068006
7778742049 1123595505617977528089887640449438202245689422109
12586269025 11235955056179775280898876404494382022469480490115
20365011074 112359550561797752808988764044943820224715169912224
32951280099 1123595505617977528089887640449438202247184650402339
53316291173 11235955056179775280898876404494382022471899820314563
86267571272 112359550561797752808988764044943820224719084470716902
139583862445 1123595505617977528089887640449438202247190984291031465
225851433717 11235955056179775280898876404494382022471910068761748367
365435296162 112359550561797752808988764044943820224719101053052779832
591286729879 1123595505617977528089887640449438202247191011121814528199
956722026041 11235955056179775280898876404494382022471910112174867308031
1548008755920 112359550561797752808988764044943820224719101123296681836230
2504730781961 1123595505617977528089887640449438202247191011235471549144261
4052739537881 11235955056179775280898876404494382022471910112358768230980491
6557470319842 112359550561797752808988764044943820224719101123594239780124752
10610209857723 1123595505617977528089887640449438202247191011235953008011105243
17167680177565 11235955056179775280898876404494382022471910112359547247791229995
27777890035288 112359550561797752808988764044943820224719101123595500255802335238
44945570212853 1123595505617977528089887640449438202247191011235955047503593565233
72723460248141 11235955056179775280898876404494382022471910112359550547759395900471
117669030460994 112359550561797752808988764044943820224719101123595505595262989465704
190392490709135 1123595505617977528089887640449438202247191011235955056143022385366175
1 kill "fibnos.txt"
5 open "fibnos.txt" for output as #2
10 Fiba=1:Fibb=1:N=1
20 for Lvl=1 to 100
30 H=Fiba:Fiba=Fibb:Fibb=Fibb+H
40 N=N*10+Fiba
50 print Fiba,N:print #2,Fiba,N
60 next
70 close #2
The output list was truncated so as not to widen the area of the comment.
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Posted by Charlie
on 2015-07-12 08:52:04 |
The Fibonacci connection
