MathDB

Let

a

b

and

k

be three positive integers. We define two sequences

\left( a_{n}\right)

and

\left( b_{n}\right)

by the starting values a_{1}\equal{}a and b_{1}\equal{}b and the recurrent equations a_{n\plus{}1}\equal{}ka_{n}\plus{}b_{n} and b_{n\plus{}1}\equal{}kb_{n}\plus{}a_{n} for each positive integer

n

. Prove that if

a_{1}\perp b_{1}

a_{2}\perp b_{2}

and

a_{3}\perp b_{3}

hold, then

a_{n}\perp b_{n}

holds for every positive integer

n

. Here, the abbreviation

x\perp y

stands for "the numbers

x

and

y

are coprime".

Problem Statement