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GCD of consecutive terms

Source:

March 14, 2006

number theorygreatest common divisoralgorithmalgebrapolynomialEuclidean algorithm

Problem Statement

The numbers in the sequence 101, 104, 109, 116,

\dots

are of the form

a_n = 100 + n^2

, where

n = 1

, 2, 3,

\dots

. For each

n

, let

d_n

be the greatest common divisor of

a_n

and

a_{n + 1}

. Find the maximum value of

d_n

n

ranges through the positive integers.