Exercise 5.5.2 (Find $c$ such that the equation $-7x^2 + 15y^2 + cz^2 = 0$ has a nontrivial integral solution)

Question

What is the least positive square-free integer $c$ such that $(c,105) = 1$, and such that the equation $-7x^2 + 15y^2 + cz^2 = 0$ has a nontrivial integral solution.

Richard Ganaye · Accepted Answer

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

\begin{proof} Put $a = -7, b = 15$. If $c = 13$, then $c \wedge a = c \wedge b = 1$, so $c \wedge 105 = 1$, thus $abc$ is square-free.
\begin{itemize}
\item $-bc = -195 \equiv 1^2 \pmod 7$,
\item $-ca = 91 \equiv 1^2 \pmod {15}$,
\item $-ab = 105 \equiv 1^2 \pmod {13}$,
\end{itemize}
then Lagrange's theorem shows that $-7x^2 + 15y^2 + cz^2 = 0$ has a nontrivial integral solution (for instance $(7,3,4)$).

For no value of $c$  less that $13$, $-7x^2 + 15y^2 + cz^2 = 0$ has a nontrivial integral solution, as verified with this Sagemath program:
\begin{verbatim}
a, b = -7, 15
c = 0
not_found = True
while not_found:
    c += 1
    if gcd(c, 105) == 1 and c.is_squarefree():
        (u, v, w) = (Mod(- b * c, a), Mod(- c * a, b), Mod(- a * b, c))
        if u.is_square() and v.is_square() and w.is_square():
            not_found = False
c
          13
\end{verbatim}
(The next suitable value is $c = 73$.)
\end{proof}

Exercise 5.5.2 (Find $c$ such that the equation $-7x^2 + 15y^2 + cz^2 = 0$ has a nontrivial integral solution)

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Exercise 5.5.2 (Find $c$ such that the equation $-7x^2 + 15y^2 + cz^2 = 0$ has a nontrivial integral solution)

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