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Commit 25e7426f authored by Jan Snellman's avatar Jan Snellman
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tenta 2025

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\noindent{
Talteori 6hp, Kurskod TATA54, Provkod TEN1 \\
08 Jan 2025 kl 08 -- 12 \\
LINKPINGS UNIVERSITET\\
Matematiska Institutionen\\
Examinator: Jan Snellman}
\bigskip
\bigskip
Inga hjlpmedel.
Alla problem ger maximalt 3 pong. Full pong krver fullstndig lsning.
8p rcker fr betyg 3, 11p fr betyg 4, 14p fr betyg 5.
\bigskip
\bigskip
\begin{enumerate}[1)]
\item Primfaktorisera 444 ver de Gaussiska heltalen,
dvs skriv 444 som en produkt av Gaussiska primtal.
\item Ls kongruensen \(x^3 \equiv 12 \mod 5^{3}\).
\item Bestm kedjebrksutveckligen av \(\sqrt{13}\).
\item \((x,y)=(10,3)\) r en positiv heltalslsning till Pell's equation \(x^2-11y^2=1\).
Hitta en sdana lsning till.
\item Visa att \(\phi(n^2) = n \phi(n)\).
\item Hitta, om mjligt, ett primtal \(p \ge 7\) samt ett positivt heltal \(a\) s
att \(a\) och \(a+2\) bda r kvadratiska residyer modulo \(p\).
\end{enumerate}
\end{document}
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