Problems
Among the first \(20\) Fibonacci numbers: \(F_0 = 0,F_1 = 1,F_2 = 1, F_3 = 2, F_4 = 3,..., F_{20} = 6765\) find all numbers whose digit-sum is equal to their index. For example, \(F_1=1\) fits the description, but \(F_{20} = 6765\) does not, since \(6+7+6+5 \neq 20\).
Among the first \(20\) Fibonacci numbers: \(F_0 = 0,F_1 = 1,F_2 = 1, F_3 = 2, F_4 = 3,..., F_{20} = 6765\) check whether the numbers with prime index are prime. The index is another name for a number’s place in the sequence.
Consider Pascal’s triangle: it starts with \(1\), then each entry in the triangle is the sum of the two numbers above it. Prove that the diagonals of Pascal’s triangle add up to Fibonacci numbers.
Prove for any integers \(m,n\ge0\) that \(F_{m+n} = F_{m-1}F_n + F_mF_{n+1}\).
Corollary: if \(k\mid n\), then \(F_k\mid F_n\). This can be proven by induction if we write \(n=sk\) for a natural \(s\), then \[F_{k+(s-1)k} = F_{k-1}F_{(s-1)k} + F_kF_{(s-1)k+1}.\]
Denote by \(\gcd(m,n)\) the greatest common divisor of numbers \(m,n\), namely the largest possible \(d\) which divides both \(n\) and \(m\). Prove for any \(m,n\) that \[\gcd(F_n,F_m) = F_{\gcd(m,n)}.\]
What’s the sum of the Fibonacci numbers \(F_0+F_1+F_2+...+F_n\)?
What’s the sum \(\frac{F_2}{F_1}+\frac{F_4}{F_2}+\frac{F_6}{F_3}+...+\frac{F_{18}}{F_9}+\frac{F_{20}}{F_{10}}\)?