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:

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

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


Sources

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