Irrationally, Can This Happen?

True or False?

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


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

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