Difference between Two Squares equal to Repunit/Examples/R 4
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Example of Difference between Two Squares equal to Repunit
We have that:
\(\ds 1111\) | \(=\) | \(\ds 101 \times 11\) | ||||||||||||
\(\ds 1111\) | \(=\) | \(\ds 1 \times 1111\) |
\(\ds 1111\) | \(=\) | \(\ds 1111 \times 1\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {1111 + 1} 2\) | \(=\) | \(\ds 556\) | |||||||||||
\(\ds \frac {1111 - 1} 2\) | \(=\) | \(\ds 555\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 556^2 - 555^2\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 309 \, 136 - 308 \, 025\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1111\) |
\(\ds 1111\) | \(=\) | \(\ds 101 \times 11\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {101 + 11} 2\) | \(=\) | \(\ds 56\) | |||||||||||
\(\ds \frac {101 - 11} 2\) | \(=\) | \(\ds 45\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 56^2 - 45^2\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 3136 - 2025\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1111\) |
$\blacksquare$
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
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $2025$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $2025$