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

Source: Serbia Math Olympiad 2016 P6

April 2, 2016

number theorynumber theory unsolved

Problem Statement

Let

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

.

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