TITLE: Are there infinitely many palindromes whose sum is a perfect square?
QUESTION [5 upvotes]: I am just curious on wether there are infinitely many palindromes say $p_1$ and $p_2$ satisfying:
$p_1^2+p_2^2$ is a perfect square with $\gcd(p_1,p_2)=1$.
I believe that there are some but, are there infinitely many of them?
Thanks for your help.

REPLY [6 votes]: Yes, there are infinitely many. For any $n\in\mathbb{N}$ let
$$\begin{align}
a_n&=3+4\times10^{n}+7\times10^{2n}+4\times10^{3n}+3\times10^{4n}\\
b_n&=4+2\times10^{n}+8\times10^{2n}+2\times10^{3n}+4\times10^{4n}\\
c_n&=5+4\times10^{n}+11\times10^{2n}+4\times10^{3n}+5\times10^{4n}
\end{align}$$
Then $a_n$ and $b_n$ are palindromes and 
$$
a_n^2+b_n^2=c_n^2.
$$
Moreover, $\gcd(a_n,b_n)=1$. Let
$$\begin{align}
A_n&=7+8\times10^{n}+8\times10^{2n}+8\times10^{3n}\\
B_n&=-\frac12\bigl(10+21\times10^{n}+22\times10^{2n}+12\times10^{3n}\bigr)
\end{align}$$
Then
$$
A_n\,a_n+B_n\,b_n=1.
$$
I found this identity doing a brute force search. Any palindrome with an even number of digits is divisible by $11$, so one (or both) of the $p_i$ must have an odd number of digits. I have searched for $p_1$ with $2\,k+1$ digits, $1\le j\le 7$. These are the results. Included are also some examples of $p_1$ with $17$ digits.
$$
\begin{array}{ll}
 313 & 48984 \\
 464 & 777 \\
 25652 & 55755 \\
 34743 & 42824 \\
 52625 & 80808 \\
 80308 & 5578755 \\
 2152512 & 575575 \\
 2532352 & 5853585 \\
 5679765 & 23711732 \\
 304070403 & 402080204 \\
 341484143 & 420282024 \\
 345696543 & 422282224 \\
 355949553 & 690019910096 \\
 359575953 & 401141104 \\
 27280108272 & 55873637855 \\
 3004007004003 & 4002008002004 \\
 3044529254403 & 4022208022204 \\
 3410048400143 & 4200028200024 \\
 3414249424143 & 4202028202024 \\
 3450569650543 & 4224448444224 \\
 6381414141836 & 778233332877 \\
 395734505437593 & 426982282289624 \\
 404990565099404 & 747709181907747 \\
 461781161187164 & 778676101676877 \\
 30004000700040003 & 40002000800020004 \\
 30040410801404003 & 40020200800202004 \\
 30044412921444003 & 40022200800222004 \\
 30081842624818003 & 40041401210414004 \\
 30401040804010403 & 40200020802000204 \\
 30405060906050403 & 40202020802020204 \\
 32682698889628623 & 46020004840002064 \\
 34100004840000143 & 42000002820000024 \\
 34104204940240143 & 42002002820020024 \\
 34140434943404143 & 42024404840442024 \\
 34505056965050543 & 42244646864644224 \\
\end{array}
$$