Fibonacci solution..

Question

Let $\ p(n)$ be a function which follows fibonacci sequence. Domain of function is only natural numbers. Now is there any way to figure out number of solutions for the equation $\ p(n) = n+1$ analytically?

Answer 1

Let $p(n)=F(n)$ and $F(0)=0, F(1)=1, F(n+2)=F(n+1)+F(n)$.

Note: $$\begin{align}F(0)=&0<0+1;\\ F(1)=&1<1+1;\\ F(2)=&1<2+1;\\ F(3)=&2<3+1;\\ F(4)=&3<4+1;\\ F(5)=&5<5+1;\\ F(6)=&8>6+1;\\ F(7)=&13>7+1;\\ F(n)=&F(n)>n+1,n\ge 6.\end{align}$$

Answer 2

HINT: From Binet's formula we know, that $p(n)$ behaves as $(\frac{1+\sqrt{5}}{2})^n\frac1{\sqrt5}$. Hence there are no large solutions. The rest we can check manually.