table of some fundamental units
SomethingRelatedToFundamentalUnits
2008-04-07 14:32:08
2008-04-08 16:47:21
Result
fundamental units
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Below, we tabulate the fundamental units $\eta$ of first real quadratic fields $\mathbb{Q}(\sqrt{d})$; the number $\omega$ is $\displaystyle\frac{1\!+\!\sqrt{d}}{2}$ for\,
$d \equiv 1 \pmod{4}$\, and $\sqrt{d}$ for\, $d \not\equiv 1 \pmod{4}$.
\begin{center}
\begin{tabular}{||c|c||c|c||}
\hline\hline
$d$ & $\eta$ & $d$ & $\eta$\\
\hline\hline
$2$ & $1+\omega$ & $47$ & $48+7\omega$\\
\hline
$3$ & $2+\omega$ & $51$ & $50+7\omega$\\
\hline
$5$ & $\omega$ & $53$ & $3+\omega$\\
\hline
$6$ & $5+2\omega$ & $55$ & $89+12\omega$\\
\hline
$7$ & $8+3\omega$ & $57$ & $131+40\omega$\\
\hline
$10$ & $3+\omega$ & $58$ & $99+13\omega$\\
\hline
$11$ & $10+3\omega$ & $59$ & $530+69\omega$\\
\hline
$13$ & $1+\omega$ & $61$ & $17+5\omega$\\
\hline
$14$ & $15+4\omega$ & $62$ & $63+8\omega$\\
\hline
$15$ & $4+\omega$ & $65$ & $7+2\omega$\\
\hline
$17$ & $3+2\omega$ & $66$ & $65+8\omega$\\
\hline
$19$ & $170+39\omega$ & $67$ & $48842+5967\omega$\\
\hline
$21$ & $2+\omega$ & $69$ & $11+3\omega$\\
\hline
$22$ & $197+42\omega$ & $70$ & $251+30\omega$\\
\hline
$23$ & $24+5\omega$ & $71$ & $3480+413\omega$\\
\hline
$26$ & $5+\omega$ & $73$ & $943+250\omega$\\
\hline
$29$ & $2+\omega$ & $74$ & $43+5\omega$\\
\hline
$30$ & $11+2\omega$ & $77$ & $4+\omega$\\
\hline
$31$ & $1520+273\omega$ & $78$ & $53+6\omega$\\
\hline
$33$ & $19+8\omega$ & 79 & $80+9\omega$\\
\hline
$34$ & $35+6\omega$ & $82$ & $9+\omega$\\
\hline
$35$ & $6+\omega$ & $83$ & $82+9\omega$\\
\hline
$37$ & $5+2\omega$ & $85$ & $4+\omega$\\
\hline
$38$ & $37+6\omega$ & $86$ & $10405+1122\omega$\\
\hline
$39$ & $25+4\omega$ & $87$ & $28+3\omega$\\
\hline
$41$ & $27+10\omega$ & $89$ & $447+106\omega$\\
\hline
$42$ & $13+2\omega$ & $91$ & $1574+165\omega$\\
\hline
$43$ & $3482+531\omega$ & $93$ & $13+3\omega$\\
\hline
$46$ & $24335+3588\omega$ & $94$ & $2143295+221064\omega$\\
\hline
\end{tabular}
\end{center}
\begin{thebibliography}{9}
\bibitem{BS}{\sc S. Borewicz \& I. Safarevic}: {\em Zahlentheorie}.\, Birkh\"auser Verlag. Basel und Stuttgart (1966).
\end{thebibliography}