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
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1444$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1681$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $49$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1444$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1681$