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I came across very interesting sequence based on fibonacci sequence. From fibonacci numbers we choose only elements with digit sum=index in fibonacci sequence. It is very interesting that we most probably know the greatest number with that property. How come we can say probably I understand that sequence is growing very quickly, but what is the idea of probably knowing the greatest number. How did mathematicans conclude that? Thanks!

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To quote this webpage, which further quotes Robert Dawson:

Robert Dawson of Saint Mary's University, Nova Scotia, Canada summarises a simple statistical argument (originally in the article referred to below by David Terr) that suggests there may be only a finite number (in fact, just 20 numbers) in this series:

"The number of decimal digits in Fib(N) can be shown to be about 0.2 N, and the average value of a decimal digit is (0+1+...+8+9)/10 = 4·5. Thus, unless the digits of Fibonacci numbers have some so-far undiscovered pattern, we would expect the digit sum to be about 0.9 N. This falls further behind N as N gets larger. Fib(2222) (with 465 digits) is the largest known Fibonacci number with this property. There are no others with N<5000, and it seems likely that Fib(2222) is actually the largest one. However, no proof exists!"

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