The sequence {S(n)} is defined by the relationship: S(n) = 100 + n², whenever n is a positive integer.

If G(n) = gcd(S(n), S(n+1)), then find the maximum value of G(n).

If G(n) = gcd(S(n), S(n+1)), then it divides their difference, which is simply (2n+1).

So, in the example given, (n^2+100)/(2n+1) is an integer.

Generalising: (n^2+k)/(2n+1) must also be an integer.

We can always make the substitution: ((2k)^2+k)/(2*(2k)+1) to return the integer, k. Since in the example given, k=100, the prime 401 must divide all solutions {S(n), S(n+1)}

And so on.

Posted by broll on 2014-10-12 21:57:18