Difference between Two Squares equal to Repunit/Examples/R 7
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Example of Difference between Two Squares equal to Repunit
We have that:
| \(\ds 1 \, 111 \, 111\) | \(=\) | \(\ds 239 \times 4649\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1 \times 1 \, 111 \, 111\) |
| \(\ds 1 \, 111 \, 111\) | \(=\) | \(\ds 1 \, 111 \, 111 \times 1\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {1 \, 111 \, 111 + 1} 2\) | \(=\) | \(\ds 555 \, 556\) | |||||||||||
| \(\ds \frac {1 \, 111 \, 111 - 1} 2\) | \(=\) | \(\ds 555 \, 555\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 555 \, 556^2 - 555 \, 555^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 308 \, 642 \, 469 \, 136 - 308 \, 641 \, 358 \, 025\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1 \, 111 \, 111\) |
| \(\ds 1 \, 111 \, 111\) | \(=\) | \(\ds 4649 \times 239\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {4649 + 239} 2\) | \(=\) | \(\ds 2444\) | |||||||||||
| \(\ds \frac {4649 - 239} 2\) | \(=\) | \(\ds 2205\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 2444^2 - 2205^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 5 \, 973 \, 136 - 4 \, 862 \, 025\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1 \, 111 \, 111\) |
$\blacksquare$
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