Let a,b be two distinct odd natural numbers.Define a Sequence <an>n≥0 like following:
a_1 \equal{} a \\
a_2 \equal{} b \\
a_n \equal{} \text{largest odd divisor of }(a_{n \minus{} 1} \plus{} a_{n \minus{} 2}).
Prove that there exists a natural number N such that a_n \equal{} gcd(a,b) \forall n\ge N. AMCUSA(J)MOUSAMOnumber theory proposednumber theory