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!
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!"