Difference between Two Squares equal to Repunit/Examples/R 5
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Example of Difference between Two Squares equal to Repunit
We have that:
| \(\ds 11 \, 111\) | \(=\) | \(\ds 41 \times 271\) | ||||||||||||
| \(\ds 11 \, 111\) | \(=\) | \(\ds 1 \times 11 \, 111\) |
| \(\ds 11 \, 111\) | \(=\) | \(\ds 11 \, 111 \times 1\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {11 \, 111 + 1} 2\) | \(=\) | \(\ds 5556\) | |||||||||||
| \(\ds \frac {11 \, 111 - 1} 2\) | \(=\) | \(\ds 5555\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 5556^2 - 5555^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 30 \, 869 \, 136 - 30 \, 858 \, 025\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 11 \, 111\) |
| \(\ds 11 \, 111\) | \(=\) | \(\ds 271 \times 41\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {271 + 41} 2\) | \(=\) | \(\ds 156\) | |||||||||||
| \(\ds \frac {271 - 41} 2\) | \(=\) | \(\ds 115\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 156^2 - 115^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 24 \, 336 - 13 \, 225\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 11 \, 111\) |
$\blacksquare$
Sources
- Dec. 1986: C.B. Lacampagne and J.L. Selfridge: Pairs of Squares with Consecutive Digits (Math. Mag. Vol. 59, no. 5: pp. 270 – 275) www.jstor.org/stable/2689401