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}.
As the numbers get progressively larger, one can see that there is a pattern of primes exponetially increasing and multiplied by new primes. For example:
a_{1} = 19
a_{3} = 19*2^{1}*3^{1}
a_{5} = 19*2^{3}*3^{3}
a_{7} = 19*2^{4}*3^{4}*211^{1}
and
a_{2} = 95
a_{4} = 95*7^{1}
a_{6} = 95*7^{2}*31^{1}
a_{8} = 95*7^{3}*31^{2}*7561^{1}
It is possible that in the progression a_{n} and a_{n+1} will eventually share a common prime other than 19. Yet without the aid of a computer program, I woud find it difficult to find these common factors. Thus, my initial guess for the GCD is the known common factor between 19 and 95 [95=19*5]  that is, 19.
Edited on August 20, 2006, 5:25 am

Posted by Dej Mar
on 20060819 14:01:30 