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1 2 $begingroup$ Let $N in mathbb{N}$ not a square, show that the continued fraction expansion of $sqrt{N}/lfloorsqrt{N}rfloor$ is $[1,overline{a_1,a_2,dots,2}]$ . My notations: the fractional part of $a$ is denoted by ${a}$ . Let $N_1 < N < N_2$ , where $N_1$ and $N_2$ are squares the closest to $N$ . $sqrt{N}/lfloorsqrt{N}rfloor = (sqrt{N} - lfloorsqrt{N}rfloor+lfloorsqrt{N}rfloor)/lfloorsqrt{N}rfloor = 1 + (sqrt{N} - lfloorsqrt{N}rfloor)/lfloorsqrt{N}rfloor$ . Now I'm stuck at the further steps. number-theory elementary-number-theory continued-fractions share | cite | improve this question edited Dec 15 '18 at 16:45