MathDB

2

Part of 2015 Silk Road

Problems(1)

(a_n^2+b_n^2)(a_{n+1}^2+b_{n+1}^2), (a_n^2+a_{n+1}^2)(b_n^2+b_{n+1}^2)

Source: SRMC 2015

9/2/2018
Let {an}n≥1\left\{ {{a}_{n}} \right\}_{n \geq 1} and {bn}n≥1\left\{ {{b}_{n}} \right\}_{n \geq 1} be two infinite arithmetic progressions, each of which the first term and the difference are mutually prime natural numbers. It is known that for any natural nn, at least one of the numbers (an2+an+12)(bn2+bn+12)\left( a_n^2+a_{n+1}^2 \right)\left( b_n^2+b_{n+1}^2 \right) or (an2+bn2)(an+12+bn+12)\left( a_n^2+b_n^2 \right) \left( a_{n+1}^2+b_{n+1}^2 \right) is an perfect square. Prove that an=bn{{a}_{n}}={{b}_{n}}, for any natural nn .
Perfect SquareArithmetic Progressionnumber theorySequencesSequence