For-each-positive-integer-n-define-a-n-30-n-2-and-d-n-gcd-a-n-a-n-1-Find-the-set-of-all-values-that-are-taken-by-d-n-and-show-by-examples-that-each-of-these-values-are-attained-

Question Number 22625 by Rasheed.Sindhi last updated on 21/Oct/17

For each positive integer n define

a_{n} = 30 + n^{2}, and d_{n} = \gcd (a_{n}, a_{n + 1}) .

Find the set of all values that are

taken by d_{n} and show by examples

that each of these values are attained .

Commented by Tinkutara last updated on 21/Oct/17

P r o b a b l y d_{n} = {1, 11, 121}

Commented by Rasheed.Sindhi last updated on 21/Oct/17

Are you sure that 11^{3} = 1331

or any other number {doesn}^{'} t

include ? Or even the above set

is finite ? Can we prove it ?

In general if a_{n} = k + n^{2}, for what values

of k, d_{n} is finite ?