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.


11A55 Continued fractions
11R11 Quadratic extensions
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