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\hypersetup{
  pdftitle={A Congruence Obstruction to Ibn al-Khawwam's Eleventh Problem},
  pdfauthor={Will Howe, Grant Huffman, Jensen Kohlmeyer, Liam Kruer},
  pdfsubject={A proof of the nonexistence of rational solutions to Ibn al-Khawwam's eleventh problem},
  pdfkeywords={Diophantine equations, rational squares, congruences, Ibn al-Khawwam}
}

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\title[Ibn al-Khaww\={a}m's eleventh problem]
  {A congruence obstruction to Ibn al-Khaww\={a}m's eleventh problem}

\author{Will Howe}
\email{howew@purdue.edu}
\author{Grant Huffman}
\email{huffma97@purdue.edu}
\author{Jensen Kohlmeyer}
\email{jenwin@purdue.edu}
\author{Liam Kruer}
\email{kruerl@purdue.edu}

\date{November 11, 2026}
\subjclass[2020]{11D09, 11A07}
\keywords{Diophantine equations, rational squares, congruences,
  Ibn al-Khaww\={a}m}

\begin{document}

\begin{abstract}
We prove that there are no two rational squares whose sum is ten and
such that adding its own square root to each gives another rational
square. This is the existence question in Ibn al-Khaww\={a}m's eleventh
arithmetic problem. The proof uses only the residues of squares modulo
three and a divisibility argument.
\end{abstract}

\maketitle

\section{The problem and its solution}

Ibn al-Khaww\={a}m's eleventh problem asks for a decomposition of ten
into two rational squares, each remaining a rational square after
addition of its positive square root. The problem occurs in the
unsolved-problems section of his thirteenth-century arithmetic treatise.
Its Arabic text and equations are reproduced by
Fazl\i o\u glu~\cite[pp.~103--104, 123--124]{Fazlioglu}.
It also appears in Derouiche's list~\cite[\S 5.4]{Derouiche}.
Writing the original squares as $x^2$ and $y^2$ gives
\begin{equation}\label{eq:rational}
  x^2+y^2=10,\qquad x^2+x=u^2,\qquad y^2+y=v^2.
\end{equation}
These are the same equations used by Rashed~\cite[p.~143]{Rashed}.
The following theorem settles the existence question even when all four
variables are allowed to have either sign.

\begin{theorem}\label{thm:main}
The system~\eqref{eq:rational} has no solution in $\mathbb{Q}^4$.
\end{theorem}

\begin{proof}
Suppose that a rational solution exists. Write
\[
  x=\frac{a}{c},\qquad y=\frac{b}{c},\qquad
  u=\frac{d}{c},\qquad v=\frac{e}{c}.
\]
with $a,b,c,d,e\in\mathbb Z$, $c>0$, and $\gcd(a,b,c,d,e)=1$.
Clearing denominators gives
\begin{equation}\label{eq:integer}
  a^2+b^2=10c^2,\qquad d^2=a(a+c),\qquad e^2=b(b+c).
\end{equation}

First, $3\nmid c$. Indeed, if $3\mid c$, then the first equation
of~\eqref{eq:integer}, together with the fact that a square is congruent
to $0$ or $1$ modulo $3$, implies $3\mid a$ and $3\mid b$.
The remaining equations then imply $3\mid d$ and $3\mid e$.
This contradicts $\gcd(a,b,c,d,e)=1$.

Consequently $c^2\equiv1\pmod3$, and the first equation
of~\eqref{eq:integer} implies that exactly one of $a,b$ is divisible
by $3$. Interchanging $(a,d)$ and $(b,e)$ if necessary, assume $3\mid b$.
Then $3\mid e$, so $9\mid e^2=b(b+c)$. Since $3\nmid b+c$, it follows
that
\begin{equation}\label{eq:divisibility}
  9\mid b.
\end{equation}

\newpage
Both $a$ and $c$ are nonzero modulo $3$. If $a\equiv c\pmod3$, then
$d^2=a(a+c)\equiv2c^2\equiv2\pmod3$, which is impossible.
Therefore
\begin{equation}\label{eq:sign}
  a\equiv-c\pmod3,\qquad 3\mid d.
\end{equation}

Now multiply $d^2=a(a+c)$ by $a-c$ and use the first equation
of~\eqref{eq:integer}. This yields the identity
\begin{equation}\label{eq:identity}
  (a-c)d^2=a(a^2-c^2)=a(9c^2-b^2).
\end{equation}
By~\eqref{eq:divisibility} and~\eqref{eq:sign}, write $b=9k$ and
$d=3\ell$ with $k,\ell\in\mathbb Z$. Dividing~\eqref{eq:identity}
by $9$ gives
\[
  (a-c)\ell^2=a(c^2-9k^2).
\]
Modulo $3$, the congruence $a\equiv-c$ implies $a-c\equiv c$.
Thus
\[
  c\ell^2\equiv-c^3\pmod3.
\]
Since $3\nmid c$, cancellation gives
\[
  \ell^2\equiv-c^2\equiv-1\equiv2\pmod3,
\]
contradicting the possible residues of an integer square.
\end{proof}

\section*{Acknowledgment of AI assistance}
OpenAI Codex assisted with developing and checking the congruence
argument and preparing the manuscript.

\begin{thebibliography}{3}

\bibitem{Derouiche}
K.~I.~A. Derouiche,
\emph{A list of open problems in number theory posed by
Ibn al-Khaww\={a}m (13th century): Historical and arithmetic analysis},
preprint (2025), \S 5.4, Problem~11.
\href{https://arxiv.org/abs/2511.08400v1}{arXiv:2511.08400v1}.

\bibitem{Fazlioglu}
\.{I}. Fazl\i o\u glu,
\emph{\.{I}bn el-Havv\^am, eserleri ve el-Fev\^aid el-Bah\^aiyye fi
el-Kav\^aid el-His\^abiyye'deki \c{c}\"oz\"ums\"uz problemler bahsi},
in \emph{Osmanl\i{} Bilimi Ara\c{s}t\i rmalar\i},
F.~G\"unerg\"un (ed.), \.{I}stanbul, 1995, pp.~103--104, 123--124.
\href{https://isamveri.org/pdfdrg/D040138/1995/1995_FAZLIOGLUI.pdf}{Critical edition and commentary}.

\bibitem{Rashed}
R. Rashed,
\emph{Histoire de l'analyse diophantienne classique:
D'Ab\={u} K\={a}mil \`a Fermat},
Scientia Graeco-Arabica, vol.~12,
De Gruyter, Berlin--Boston, 2013, p.~143.
\href{https://doi.org/10.1515/9783110337884}{doi:10.1515/9783110337884}.

\end{thebibliography}

\end{document}