Difference between Two Squares equal to Repunit/Examples/R 3
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Example of Difference between Two Squares equal to Repunit
We have that:
| \(\ds 111\) | \(=\) | \(\ds 3 \times 37\) | ||||||||||||
| \(\ds 111\) | \(=\) | \(\ds 1 \times 111\) |
| \(\ds 111\) | \(=\) | \(\ds 111 \times 1\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {111 + 1} 2\) | \(=\) | \(\ds 56\) | |||||||||||
| \(\ds \frac {111 - 1} 2\) | \(=\) | \(\ds 55\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 56^2 - 55^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 3136 - 3025\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 111\) |
| \(\ds 111\) | \(=\) | \(\ds 37 \times 3\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {37 + 3} 2\) | \(=\) | \(\ds 20\) | |||||||||||
| \(\ds \frac {37 - 3} 2\) | \(=\) | \(\ds 17\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 20^2 - 17^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 400 - 289\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 111\) |
$\blacksquare$
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