Let's look at the sequence with terms
a
_{1}=19,
a
_{2}=95, and a
_{n+2}=LCM(a
_{n+1},a
_{n})+a
_{n}
LCM stands for Least Common Multiple, and n is a positive integer.
Find the Greatest Common Divisor (GCD) of terms a_{4096} and a_{4097}.
(In reply to
re(3): Just a guess...  proof by Charlie)
OK then, I see where you are coming from. If we suppose that P is
a prime that divides both a(n+2) and a(n+1), but does not divide a(n),
then it must divide [lcm(a(n+1),a(n))/a(n)]+1 and hence cannot divide
R=lcm(a(n+1),a(n))/a(n). But P does divide R since it divides a(n+1)
and not a(n), and this contradiction then proves that P does divide
a(n) as well as a(n+1) and a(n+2).
The easily seen fact that a prime P that is contained in A but not in B must be contained in lcm(A,B)/B is the crux, then.

Posted by Richard
on 20060819 21:54:57 