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Consecutive powers in continued fractions. (English) Zbl 0764.11010
Let $$m>0$$ be a square-free integer and consider the continued fraction expansion of $$\omega=\sqrt{m}$$ (resp. $$\omega=(1+\sqrt{m})/2)$$ whenever $$m\not\equiv1\pmod 4$$ (resp. $$m\equiv 1\pmod 4$$). Then $$\omega=[a_ 0,\overline{a_ 1,\dots,a_ \ell}]$$ with period length $$\ell$$; $$a_ 0=\lfloor\omega\rfloor$$, $$a_ i=\lfloor (P_ i+\sqrt{m})/Q_ i\rfloor$$ for $$i\geq 1$$ where $$\lfloor\;\rfloor$$ denotes the greatest integer function; $$(P_ 0,Q_ 0)=(0,1)$$ (resp. (1,2)) if $$m\not\equiv 1 \pmod 4$$ (resp. $$m\equiv 1 \pmod 4$$); $$P_{i+1}=a_ i Q_ i-P_ i$$ and $$Q_{i+1}Q_ i=m-P_{i+1}^ 2$$ for $$i\geq 0$$. The authors investigate conditions for the existence of at least three consecutive $$Q_ i/Q_ 0$$’s in a row which are the power of a single integer $$a>1$$. In fact they describe the form of all such $$m$$’s, and include a simple formula for the period length. This is a long and rather technical paper, and the results are amazing and interesting; since there are so many computations, most mathematicians will be willing to accept the results without checking the details.

##### MSC:
 11A55 Continued fractions 11R11 Quadratic extensions
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