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Prove that the sequence..

Source: Indian Postal Coaching 2005 Problem 1

September 22, 2005

number theory unsolvednumber theory

Problem Statement

Consider the sequence

<{a_n}>

of natural numbers such that {i}

a_n

is a square numver for all

n

; (ii)

a_{n+1} - a_n

is either a prime or a square of a prime for each

n

. Show that

<a_n>

is a finite sequence. Determine the longest such sequence.