I was just looking up information about the legendre symbol and found it interesting that the Fibonacci numbers involve this symbol. This is what I found on wikipedia.

The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … are defined by the recurrence F1 = F2 = 1, Fn+1 = Fn + Fn-1.

If p is a prime number then

F_{p-\left(\frac{p}{5}\right)} \equiv 0 \pmod p,\;\;\; F_{p} \equiv \left(\frac{p}{5}\right) \pmod p.

For example,

(\tfrac{2}{5}) = -1, \,\, F_3 = 2, F_2=1,

(\tfrac{3}{5}) = -1, \,\, F_4 = 3,F_3=2,

(\tfrac{5}{5}) = \;\;\,0,\,\, F_5 = 5,

(\tfrac{7}{5}) = -1, \,\,F_8 = 21,\;\;F_7=13,

(\tfrac{11}{5}) = +1, F_{10} = 55, F_{11}=89.

The symbols were hard to type in the forum so if you look up the "legendre symbol" on wikipedia can see how the formula involves the symbol.

I also read and found pretty interesting that there were many other symbols that are generalization of the legendre symbol, some of these symbols include the Jacobi symbol, the Kronecker symbol, the Hilbert symbol and the Artin symbol. It is said that this was one of the first examples of homomorphism which is defined by wikipedia as, "in abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). I find it really neat that all these symbols are related but all are named differently because of the generalizations they can make.