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Author: William A. Stein
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\title{Math 168A: MIDTERM}
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\author{William Stein}
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\date{\bf Due: Wednesday, Oct 26, 2005}
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\begin{document}
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\maketitle
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\noindent{\em The problems have equal point value, and parts of multi-part
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problems are of the same value.
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{\bf You may not talk to anybody about these problems.} You are allowed
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to use your notes, computer software, books and web pages.
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}
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\begin{enumerate}
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\item Make a conjecture about the set of primes $p$
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such that $y^2 = x^3 + x$ has exactly $p+1$ points
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modulo $p$.
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You do not have to prove your conjecture.
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You might find the following table helpful:
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%
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% E = EllipticCurve([1,0])
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% t = [(p, E.Np(p)) for p in primes(100)]
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% sage: ' & '.join([str(a) for a, _ in t])
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% _7 = '2 & 3 & 5 & 7 & 11 & 13 & 17 & 19 & 23 & 29 & 31 & 37 & 41 & 43 & 47 & 53 & 59 & 61 & 67 & 71 & 73 & 79 & 83 & 89 & 97'
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%sage: ' & '.join([str(b) for _, b in t])
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% _9 = '3 & 4 & 4 & 8 & 12 & 20 & 16 & 20 & 24 & 20 & 32 & 36 & 32 & 44 & 48 & 68 & 60 & 52 & 68 & 72 & 80 & 80 & 84 & 80 & 80'
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%sage: '|'.join(['c' for _, b in t])
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%_10 = 'c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c'
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\begin{center}
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\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline
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$p$ & 2 & 3 & 5 & 7 & 11 & 13 & 17 & 19 & 23 & 29 & 31 & 37 & 41 \\\hline
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$N_p$ & 3 & 4 & 4 & 8 & 12 & 20 & 16 & 20 & 24 & 20 & 32 & 36 & 32 \\\hline
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\end{tabular}
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\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline
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$p$ & 43 & 47 & 53 & 59 & 61 & 67 & 71 & 73 & 79 & 83 & 89 & 97 \\\hline
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$N_p$ &44 & 48 & 68 & 60 & 52 & 68 & 72 & 80 & 80 & 84 & 80 & 80\\\hline
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\end{tabular}
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\end{center}
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\item The point $(1,1)$ is a solution to the equation
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\begin{equation}\label{eqn:0}
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3 + 2y + y^2 = x^2 + 2x + 3x^3.
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\end{equation}
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Find three other solutions as follows:
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\begin{enumerate}
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\item Multiply both sides of the equation by $9$
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and let $X=3x$, $Y=3y$ to obtain an equation
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of the form
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\begin{equation}\label{eqn:1}
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Y^2 + a_1 XY + a_3 Y = X^3 + a_2X^2 + a_4 X + a_6.
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\end{equation}
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\item Find the point $P$ on (\ref{eqn:1}) corresponding
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to the point $(1,1)$ on (\ref{eqn:0}).
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\item Use the group operation on (\ref{eqn:1}), e.g., via
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SAGE, to compute several multiples of $P$.
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\item Use the multiples of $P$ to recover points on (\ref{eqn:0}).
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\end{enumerate}
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\item The right triangle with rational side
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lengths $a=24/5$, $b=35/12$, $c=337/60$ has area $7$.
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Find another right triangle with rational side lengths
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and area $7$ as follows (note that none of the sides
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lengths of this second triangle are allowed to be the same
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as the side lengths of the above triangle):
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\begin{enumerate}
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\item Find the point $P$ on $y^2 = x^3 - 49x$ corresponding
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to the triangle $(24/5, 35/12, 337/60)$ under the bijection
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from class (see notes from 2005-10-17).
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\item Compute $Q = P+P$ using the elliptic curve group law.
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\item Find the triangle corresponding to $Q$ under
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the bijection from class.
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\end{enumerate}
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\end{enumerate}
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\end{document}
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