MathDB

a) Show that there exists exactly a sequence

\left( x_n,y_n \right)_{n\ge 0}

of pairs of nonnegative integers, that satisfy the property that

\left( 1+\sqrt 33 \right)^n=x_n+y_n\sqrt 33,

for all nonegative integers

n.

b) Having in mind the sequence from a), prove that, for any natural prime

p,

at least one of the numbers

y_{p-1} ,y_p

and

y_{p+1}

are divisible by

p.

Problem Statement