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\begin{center}
\vskip 1cm{\LARGE\bf 
Four Biquadrates Whose Sum \\
\vskip 0.1in
is a Perfect Square
}
\vskip 1cm
\large
Ajai Choudhry \\
13/4 A Clay Square\\
Lucknow 226001\\
India\\
\href{mailto:ajaic203@yahoo.com}{\tt ajaic203@yahoo.com} \\
\end{center}

\vskip .2 in

\begin{abstract}
All known  nontrivial parametric solutions of  the diophantine equation
$$x_1^4+x_2^4+x_3^4+x_4^4=y^2$$ give the values of $x_i, i=1, \ldots,
4$, in terms of polynomials in two parameters. In the three simplest
solutions, these polynomials are of degrees 2, 4 and 12 respectively. In
this paper we obtain infinitely many solutions of  the equation under
consideration in terms of quadratic polynomials in two parameters, as well
as a solution  in terms of quartic polynomials in three parameters. We
also show how more general multi-parameter solutions of the equation
may be obtained.
\end{abstract}

\section{Introduction}\label{Intro}  
This paper is concerned with the problem of finding four biquadrates whose sum is a perfect square, that is, we are required to find   four integers $x_i, i=1, \ldots, 4$, such that 
\begin{equation}
x_1^4+x_2^4+x_3^4+x_4^4=y^2. \label{quarteqn}
\end{equation}
where $y$ is an integer. If we take three of the integers $x_i, i=1, \ldots, 4$, as $0$, we immediately get a trivial solution of Eq.~\eqref{quarteqn}. All other solutions of the diophantine equation \eqref{quarteqn} will be  considered as nontrivial.

All known nontrivial parametric solutions of Eq.~\eqref{quarteqn} are expressed in terms of polynomials in two parameters---the simplest being given by the identity,
\begin{equation*}
(a^4+b^4)^2=a^8+(ab)^4+(ab)^4+b^8.
\end{equation*}
Dickson \cite[pp.\  657--658]{Di}) has mentioned two other solutions---found by Fauquembergue---in which the values of $x_i,  i=1, \ldots, 4$, are given by polynomials of degrees 4 and 12 in terms of two parameters.  Recently Alvarado and Delorme \cite{AD} have obtained two solutions in which $x_i,  i=1, \ldots, 4$, are given by polynomials of degrees 16 and 28, and they have shown how infinitely many such solutions may be obtained by using elliptic curves.

In this paper we will obtain infinitely many solutions of Eq.~\eqref{quarteqn} in terms of quadratic polynomials in two parameters, as well as a solution  in terms of quartic polynomials in three parameters. We also show how more general multi-parameter solutions of the equation may be obtained.  

\section{Four biquadrates whose sum is a square}\label{parmsolutions} 

\subsection{Solutions in terms of quadratic polynomials in two parameters}\label{sol1}
In this section we will obtain solutions of the diophantine equation \eqref{quarteqn} in terms of polynomials of degree 2 in two parameters by using the 
well-known composition of forms identity,
\begin{equation}
(f^2+fg+g^2)(u^2+uv+v^2)=p^2+pq+q^2, \label{compident1}
\end{equation}
where $f, g, u, v$ are arbitrary parameters and 
\begin{equation*}
p=fu-gv, \quad q=fv+gu+gv,
\end{equation*}
together with  the  identity, 
\begin{equation}
r^4+s^4+(r+s)^4=2(r^2+rs+s^2)^2, \label{ident2}
\end{equation}
where $r$ and $s$ are arbitrary parameters.

It follows from \eqref{compident1} that
\begin{equation}
\begin{aligned}
(f^2+fg+g^2)(u^2+uv+v^2)^2&amp;=(p^2+pq+q^2)(u^2+uv+v^2), \\
&amp; = r^2+rs+s^2,
\end{aligned}
\label{compident2}
\end{equation}
where
\begin{equation}
r = fu^2 - 2guv - (f + g)v^2,\quad s = gu^2 + (2f + 2g)uv + fv^2. \label{valrs}
\end{equation}

Thus, when the values of $r$ and $s$ are given by \eqref{valrs}, we get, on combining the two identities \eqref{ident2} and \eqref{compident2}, 
\begin{equation}
r^4+s^4+(r+s)^4=2(f^2+fg+g^2)^2(u^2+uv+v^2)^4, \label{ident3}
\end{equation}
and on adding $h^4(u^2+uv+v^2)^4$ to both sides of \eqref{ident3}, we get the identity,
\begin{equation}
r^4+s^4+(r+s)^4+h^4(u^2+uv+v^2)^4=(2(f^2+fg+g^2)^2+h^4)(u^2+uv+v^2)^4. \label{ident4}
\end{equation}

If we choose integer values of $f, g$, and $h$ such that 
\begin{equation}
2(f^2+fg+g^2)^2+h^4=t^2, \label{condfgh}
\end{equation}
where $t$ is some integer, it follows from \eqref{valrs} and \eqref{ident4} that a solution of \eqref{quarteqn} is given by 
\begin{equation*}
\begin{aligned}
(x_1,x_2,x_3,x_4,y)&amp;=(fu^2 - 2guv - (f + g)v^2,  gu^2 + (2f + 2g)uv + fv^2,\\
&amp; \quad \quad (f + g)u^2 + 2fuv - gv^2,  h(u^2+uv+v^2),  t(u^2+uv+v^2)^2),
\end{aligned}
\end{equation*}
where $u$ and $v$ are arbitrary parameters.

We readily found, by computer trials, the following four sets of values of  $(f, g, h, t)$  that satisfy Eq.~\eqref{condfgh}: 
\[
(2,2,1,17),\quad (2,4,7,63),\quad (2,14,17,433), \quad(4, 22, 7, 833).\]
These four solutions of Eq.~\eqref{condfgh} immediately yield four solutions of Eq.~\eqref{quarteqn} in which the values of $x_i$ are given by  quadratic polynomials in the arbitrary parameters $u$ and $v$. The  first two solutions of  Eq.~\eqref{quarteqn} thus obtained may be written explicitly as follows:
\begin{equation}
\begin{aligned}
(x_1,x_2,x_3,x_4,y)&amp;=(2u^2 - 4uv - 4v^2,     2u^2 + 8uv + 2v^2,     4u^2 + 4uv - 2v^2,  \\
 &amp; \quad \quad u^2 + uv + v^2,   17(u^2 + uv + v^2)^2), 
\end{aligned}
\label{sol1eq1}
\end{equation}
and 
\begin{equation}
\begin{aligned}
(x_1,x_2,x_3,x_4,y)&amp;=(2u^2 - 8uv - 6v^2,    4u^2 + 12uv + 2v^2,    6u^2 + 4uv - 4v^2,  \\
 &amp; \quad \quad 7(u^2 + uv + v^2),  63(u^2 + uv + v^2)^2).
\end{aligned}
\label{sol2eq1}
\end{equation}

We will now show  how  infinitely many solutions in integers of Eq.~\eqref{condfgh} may be obtained. For fixed rational numerical values of $g$ and $h$, Eq.~\eqref{condfgh} is a quartic model of an elliptic curve. If this elliptic curve is of positive rank, we can obtain infinitely many rational points  on this elliptic curve, and by appropriate scaling, we can obtain  infinitely many integer solutions of Eq.~\eqref{condfgh}. For instance, when $(g,h)=(2,1)$, we may write Eq.~\eqref{condfgh}  as
\begin{equation}
 2f^4 + 8f^3 + 24f^2 + 32f + 33 = t^2, \label{condfghex1}
\end{equation}
and one rational point on the curve  \eqref{condfghex1} is known to be $(f,t)=(2,17)$. 

The birational transformation defined by
\begin{equation}
\begin{aligned}
f &amp; = -2(145X + 17Y + 1208)/(144X-17Y+1176),\\
 t&amp; = (4913X^3 + 121584X^2 + 982600X - 6864Y + 2585088)\\
 &amp; \quad \quad \times (144X-17Y+1176)^{-2},
\end{aligned}
\label{biratft}
\end{equation}
and
\begin{equation}
\begin{aligned}
X&amp; = 2(20f^2 + 64f + 17t + 81)/(f - 2)^2, \\
Y &amp; = 4(102f^3 + 408f^2 + 72ft + 816f + 145t + 833)/(f - 2)^3,
\end{aligned}
\label{biratXY}
\end{equation}
reduces  the elliptic curve \eqref{condfghex1} to the Weierstrass form,
\begin{equation}
Y^2 = X^3 - 200X - 1088. \label{ecweier} 
\end{equation}

A reference to Cremona's elliptic curve tables \cite{Cr} shows that the rank of the elliptic curve \eqref{ecweier} is 1. In fact, using  APECS (a package written in Maple for working with elliptic curves), we readily found the rational point $P=(217/9, 2431/27)$  on the curve \eqref{ecweier}. Since the point $P$ does not have rational coordinates, it follows from the Nagell-Lutz theorem \cite[p.\ 56]{Si} on elliptic curves that $P$ is a point of infinite order. We can now find infinitely many rational points on the curve \eqref{ecweier} using the group law, and using the relations \eqref{biratft}, we can find infinitely many rational points on the curve \eqref{condfghex1}. We can now obtain infinitely many integer solutions of Eq.~\eqref{condfgh} and thus generate infinitely many parametric solutions of Eq.~\eqref{quarteqn}.

As a numerical example, the point $P$ on the curve \eqref{ecweier} generates a solution that is equivalent to the solution \eqref{sol1eq1} while the point $2P$ generates the following parametric solution of Eq.~\eqref{quarteqn}:
\begin{equation*}
\begin{aligned}
x_1&amp;=162278u^2 + 667292uv + 171368v^2, \\
x_2&amp;= 333646u^2 + 342736uv - 162278v^2, \\
x_3&amp; =171368u^2 - 324556uv - 333646v^2, \\
x_4&amp;= 166823(u^2 + uv + v^2),\\
y&amp;= 121336214273(u^2 + uv + v^2)^2,
\end{aligned}
\end{equation*}
where $u$ and $v$ are arbitrary parameters. 

\subsection{Solutions in terms of quartic polynomials in three parameters}\label{sol2} We will now obtain a solution of \eqref{quarteqn} in which the values of  $x_i,  i=1, \ldots, 4$, are given by quartic polynomials in three parameters. 

We write 
\begin{equation}
x_1=t+a,\quad x_2=t+b,\quad x_3=t+c,\quad x_4=t, \label{subsx}
 \end{equation}
when the left-hand side of Eq.~\eqref{quarteqn} becomes a quartic function of $t$ in which the coefficient of $t^4$ is a perfect square, namely $4$. Dickson \cite[p.\  639]{Di} has described a  method, originally given by Fermat,  for  making such a quartic function a perfect square. Applying this method, we find that the  left-hand side of Eq.~\eqref{quarteqn}  becomes a perfect square if we take
\begin{equation*}
t=\phi(a,b,c)/(24(a + b - c)(a - b + c)(a - b - c)), 
\end{equation*}
where
\begin{equation*}
\begin{aligned}
\phi(a,b,c)&amp;=9(a^4+b^4+c^4)-20(a^3(b+c)+b^3(c+a)+c^3(a+b))\\
&amp; \quad \quad +54(a^2b^2+b^2c^2+c^2a^2)-12abc(a+b+c).
\end{aligned}
\end{equation*}

Using the relations \eqref{subsx}, we  now  get four biquadrates whose sum is a perfect square. On appropriate scaling, we may write these  biquadrates as $x_i^4,  i=1, \ldots, 4$, with the values of $x_i$ being given by the quartic polynomials,
\begin{equation}
x_1=\psi(a,b,c),\quad x_2=\psi(b,c,a), \quad  x_3=\psi(c,a,b),\quad x_4=\phi(a,b,c),\label{sol4eq1}
\end{equation}
where
\begin{equation*}
\begin{aligned}
\psi(a,b,c)&amp;=33a^4 - 44a^3b - 44a^3c + 30a^2b^2 + 36a^2bc \\
&amp; \quad \quad + 30a^2c^2 + 4ab^3 - 36ab^2c - 36abc^2 + 4ac^3 \\
&amp; \quad \quad + 9b^4 - 20b^3c + 54b^2c^2 - 20bc^3 + 9c^4. 
\end{aligned}
\end{equation*}

On making the invertible linear transformation,
\begin{equation*}
a = (v+w)/2, \quad b = (u+w)/2, \quad  c = (u+v)/2, 
\end{equation*}
the values of $x_i$ given by \eqref{sol4eq1} may be written as
\begin{equation}
\begin{aligned}
x_1&amp;=f(u,-v,-w),\quad &amp;x_2&amp;=(-u,v,-w),\quad \\
x_3&amp;=f(-u,-v,w), \quad &amp;x_4&amp;=f(u,v,w), 
\end{aligned}
\label{sol5eq1}
\end{equation}
where $f(u, v, w)$ is defined in terms of arbitrary parameters $u, v$, and $w$ as follows:
\begin{equation*}
f(u,v,w)=2(u^4+v^4+w^4)+3(u^2v^2+v^2w^2+w^2u^2)+6uvw(u+v+w).
\end{equation*}

With the values of $x_i,  i=1, \ldots, 4$, given by \eqref{sol5eq1}, it is readily verified that $\sum_{i=1}^4x_i^4=y^2$ where
\begin{equation}
\begin{aligned}
y&amp;=8(u^8+v^8+w^8) + 24(u^6(v^2 + w^2)+v^6(w^2+u^2)+w^6(u^2+v^2)) \\
&amp; \quad \quad + 34(u^4v^4+v^4w^4+w^4u^4) + 276u^2v^2w^2(u^2+v^2+w^2). \label{valysol5}
\end{aligned}
\end{equation}

Thus a solution of Eq.~\eqref{quarteqn} is given by \eqref{sol5eq1} and \eqref{valysol5} in terms of the arbitrary parameters $u, v$, and $w$.

\subsection{Multi-parameter solutions}\label{sol3}
We will  now describe a method of obtaining   more general  solutions of Eq.~\eqref{quarteqn} in several parameters  by using the known solutions of Eq.~\eqref{quarteqn}. 

Let a solution of  Eq.~\eqref{quarteqn} be $(x_1, x_2, x_3, x_4, y)=(\alpha_1, \alpha_2, \alpha_3, \alpha_4, \beta)$. Instead of choosing the values of $x_i,  i=1, \ldots, 4$, given by \eqref{subsx}, we now write,
\begin{equation}
\begin{aligned}
 x_1&amp;=\alpha_1t_1+\gamma_1t_2,\quad x_2&amp;=\alpha_2t_1+\gamma_2t_2,\\
 x_3&amp;=\alpha_3t_1+\gamma_3t_2, \quad x_4&amp;=\alpha_4t_1+\gamma_4t_2, 
\end{aligned}
\label{subsxrev}
 \end{equation}
where $\gamma_i, i=1, \ldots, 4$, are arbitrary parameters, and proceeding as before, we again observe  that the left-hand side of Eq.~\eqref{quarteqn} becomes a quartic function of $t_1$ in which the coefficient of $t_1^4$ is a perfect square, namely $\beta^2$ and hence, as before,  we can obtain a value of $t_1$ that yields four biquadrates whose sum is a perfect square. We thus obtain more  parametric solutions of Eq.~\eqref{quarteqn}. 

We could also  choose the parameters $\gamma_i$ in \eqref{subsxrev} such the sum of the  four biquadrates $\gamma_i^4$ is a perfect square, say $\delta^2$, and now the left-hand side of Eq.~\eqref{quarteqn} becomes a binary quartic form in  $t_1$ and $t_2$ such that the coefficients of $t_1^4$ and $t_2^4$ are $\beta^2$ and $\delta^2$, respectively.  If we now write 
\begin{equation}
y=\beta t_1^2+2\left(\sum_{i=1}^4 \alpha_i^3 \gamma_i \right)t_1t_2/\beta +\delta t_2^2, \label{valysolfin}
\end{equation}
and substitute the values of $x_i,  i=1, \ldots, 4$, and $y$ given by \eqref{subsxrev} and \eqref{valysolfin}, respectively, in Eq.~\eqref{quarteqn}, we get after suitable transposition and removal of common factors, a linear equation in $t_1$ and $t_2$ whose solution is as follows:
\begin{equation}
\begin{aligned}
t_1&amp;=2\left(\sum_{i=1}^4 \alpha_i \gamma_i^3 \right)\beta^2 -2\left(\sum_{i=1}^4 \alpha_i^3 \gamma_i \right)\beta \delta,\\
t_2&amp;=\beta^3 \delta -3\left(\sum_{i=1}^4 \alpha_i^2 \gamma_i^2 \right)\beta^2  + 2\left(\sum_{i=1}^4 \alpha_i^3 \gamma_i \right)^2.
\end{aligned}
\label{valt1t2}
\end{equation}

With the values of $t_1$ and $t_2$ given by \eqref{valt1t2}, a solution of Eq.~\eqref{quarteqn} is given by \eqref{subsxrev} and \eqref{valysolfin}. Since we can choose the values of $\alpha_i, \gamma_i, i=1, \ldots, 4$, in terms of 2 or 3 parameters, the solutions of Eq.~\eqref{quarteqn}  thus obtained are in terms of 4, 5 or 6 parameters. All the multi-parameter solutions of Eq.~\eqref{quarteqn} obtained in this manner are too cumbersome to write and accordingly we do not give any of them explicitly.

It would of considerable interest to find a parametric solution of Eq.~\eqref{quarteqn} in which the parameters can be so chosen that the value of $y$  becomes a perfect square. If that can be achieved, we will obtain four biquadrates whose sum is also a biquadrate. There seems to be no simple way of finding such a parametric solution of Eq.~\eqref{quarteqn}. 

\section{Acknowledgments}
I wish to  thank the Harish-Chandra Research Institute, Prayagraj for providing me with all necessary facilities that have helped me to pursue my research work in mathematics.

\begin{thebibliography}{9}

\bibitem{AD} A. Alvarado and J.-J. Delorme,  On the diophantine equation
$x^4 + y^4 + z^4 + t^4 = w^2$, {\it J. Integer Sequences}  {\bf 17} (2014),
\href{https://cs.uwaterloo.ca/journals/JIS/VOL17/Alvarado/alva3.html}
{Article 14.11.5}.

\bibitem{Cr} J. E. Cremona, Elliptic curve data. Available at
\url{http://johncremona.github.io/ecdata/}.

\bibitem{Di} L. E. Dickson, {\it History of the Theory of Numbers,
Vol.~2}, Chelsea Publishing Company, 1952.

\bibitem{Si} J. H. Silverman and J. Tate, {\it Rational Points on Elliptic Curves}, Springer-Verlag,  1992. 

\end{thebibliography}

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\noindent 2010 {\it Mathematics Subject Classification}: 11D25.

\noindent \emph{Keywords: }
biquadrate, quartic diophantine equation. 

\bigskip
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\vspace*{+.1in}
\noindent
Received October 9 2020;
revised version received January 18 2021.  
Published in {\it Journal of Integer Sequences}, January 23 2021.

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