Lucky, happy, fortunate Fibonacci

Number Theory Level 3

What is smallest Fibonacci number which is also lucky, happy and fortunate.


  • A number is said to be "lucky" if they survived the elimination method. Start with positive integer from 1, the first elimination removes every second term of the sequence, and afterwards, using the first surviving list, removes every \(n\)-th term (where \(n\) is the least surviving integer greater than 1). The process repeats until no more terms is available to remove, and the sequence of number remains is the sequence of "lucky number"
  • A number is said to be "happy" if, when squaring all digits and sum them up, produces a new number, and when perform the process repeatedly, produces 1. For example, 79 is "happy", as \(7^2+9^2 = 130\Rightarrow1^2+3^2+0^2 = 10\Rightarrow1^2+0^2 = 1\)
  • A number is said to be "fortunate" if it is the least positive integer \(n\) such that \[n+\prod p_m\] is a prime, where \(p_m\) is the \(m\)-th positive prime number

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