consider y1=1, y2=5, with the same formula
induction: every adjacent terms will be relatively prime
proof: base casE: y3=6. true
suppose y_n, y_(n+1) are relatively prime. then lcm(y_n,y_(n+1))=y_n*y_(n+1)
y_(n+2)=y_n*y_(n+1)+y_n
by euclidean algorithm this number is not divisible by y_(n+1)
done

lemma: x_i=y_i*15
Proof: its true for i=1,2
suppose its true for i=k, k+1
then x_(k+2)=lcm(x_(k),x_(k+1))+x_k=lcm(19*y_k,19*y_(k+
1))+19*y_k=19*((y_k*y_(k+1))+y_k)=19*y_(k+2)
QED

therefore gcd(x_1995,x_1996)=19

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