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:

\(\ds 1111\) \(=\) \(\ds 101 \times 11\)
\(\ds 1111\) \(=\) \(\ds 1 \times 1111\)


\(\ds 1111\) \(=\) \(\ds 1111 \times 1\)
\(\ds \leadsto \ \ \) \(\ds \frac {1111 + 1} 2\) \(=\) \(\ds 556\)
\(\ds \frac {1111 - 1} 2\) \(=\) \(\ds 555\)
\(\ds \leadsto \ \ \) \(\ds \) \(\) \(\ds 556^2 - 555^2\)
\(\ds \) \(=\) \(\ds 309 \, 136 - 308 \, 025\)
\(\ds \) \(=\) \(\ds 1111\)


\(\ds 1111\) \(=\) \(\ds 101 \times 11\)
\(\ds \leadsto \ \ \) \(\ds \frac {101 + 11} 2\) \(=\) \(\ds 56\)
\(\ds \frac {101 - 11} 2\) \(=\) \(\ds 45\)
\(\ds \leadsto \ \ \) \(\ds \) \(\) \(\ds 56^2 - 45^2\)
\(\ds \) \(=\) \(\ds 3136 - 2025\)
\(\ds \) \(=\) \(\ds 1111\)

$\blacksquare$


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