Irrationally, Can This Happen?

True or False?

The infinite sequence 1×2017, 2×2017, 3×2017, 4×2017, \big\lfloor 1\times \sqrt{ 2017 } \big\rfloor,\ \big\lfloor 2 \times \sqrt{ 2017 } \big\rfloor,\ \big\lfloor 3 \times \sqrt{ 2017 } \big\rfloor,\ \big\lfloor 4 \times \sqrt{ 2017 } \big\rfloor,\ \ldots contains infinitely many perfect squares.


Notation: \lfloor \cdot \rfloor denotes the floor function.

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