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 a4096 and a4097.
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:
a1 = 19
a3 = 19*21*31
a5 = 19*23*33
a7 = 19*24*34*2111
and
a2 = 95
a4 = 95*71
a6 = 95*72*311
a8 = 95*73*312*75611
It is possible that in the progression an and an+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
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Posted by Dej Mar
on 2006-08-19 14:01:30 |
Just a guess...
