6

Part of 2016 Serbia National Math Olympiad

Problems(1)

A lot of squares. Can all of them be different?

Source: Serbia Math Olympiad 2016 P6

a_1, a_2, \dots, a_{2^{2016}}

be positive integers not bigger than

2016

. We know that for each

n \leq 2^{2016}

a_1a_2 \dots a_{n} +1

is a perfect square. Prove that for some

i

a_i=1

number theorynumber theory unsolved