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 Beginning and End (Posted on 2015-07-11)
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.

 No Solution Yet Submitted by K Sengupta Rating: 3.0000 (1 votes)

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 The Fibonacci connection Comment 6 of 6 |
(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.

 Posted by Charlie on 2015-07-12 08:52:04

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