Consider the sequence $\{a_n\}_{n\in\Bbb N}$ where $a_n = \lfloor n \sqrt2 \rfloor + \lfloor n \sqrt3 \rfloor $. Show that there exists infinitely many odd and even integers in this sequence.
Infinitely many odd and even integers in a sequence.
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elementary-number-theory
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0Hint: The parity of $a_n$ is the same as $b_n=\lfloor n \sqrt3 \rfloor - \lfloor n \sqrt2 \rfloor$. Now think about the possible values of $b_{n+1}-b_n$. – 2016-04-13