Squares whose Digits can be Separated into 2 other Squares

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Theorem

The decimal representation of the following square numbers can be split into two parts which are each themselves square:

\(\displaystyle 7^2\) \(=\) \(\displaystyle 49\) $4 = 2^2$, $9 = 3^2$
\(\displaystyle 13^2\) \(=\) \(\displaystyle 169\) $16 = 4^2$, $9 = 3^2$
\(\displaystyle 19^2\) \(=\) \(\displaystyle 361\) $36 = 6^2$, $1 = 1^2$
\(\displaystyle 35^2\) \(=\) \(\displaystyle 1225\) $1 = 1^2$, $225 = 15^2$
\(\displaystyle 38^2\) \(=\) \(\displaystyle 1444\) $144 = 12^2$, $4 = 2^2$
\(\displaystyle 41^2\) \(=\) \(\displaystyle 1681\) $16 = 4^2$, $81 = 9^2$
\(\displaystyle 57^2\) \(=\) \(\displaystyle 3249\) $324 = 18^2$, $9 = 3^2$
\(\displaystyle 65^2\) \(=\) \(\displaystyle 4225\) $4 = 2^2$, $225 = 15^2$
\(\displaystyle 70^2\) \(=\) \(\displaystyle 4900\) $4 = 2^2$, $900 = 30^2$
\(\displaystyle 125^2\) \(=\) \(\displaystyle 15 \, 625\) $1 = 1^2$, $5625 = 75^2$
\(\displaystyle 130^2\) \(=\) \(\displaystyle 16 \, 900\) $16 = 4^2$, $900 = 30^2$
\(\displaystyle 190^2\) \(=\) \(\displaystyle 36 \, 100\) $36 = 6^2$, $100 = 10^2$
\(\displaystyle 205^2\) \(=\) \(\displaystyle 42 \, 025\) $4 = 2^2$, $2025 = 45^2$
\(\displaystyle 223^2\) \(=\) \(\displaystyle 49 \, 729\) $49 = 7^2$, $729 = 27^2$

This sequence is A048375 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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