Thabit Pair/Examples/220-284

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Example of Thabit Pair

$220$ and $284$ form a Thabit pair.


Proof

From Amicable Pair: $220$ and $284$:

$\sigma \left({220}\right) = \sigma \left({284}\right) = 504 = 220 + 284$

and so by definition are amicable.


Let $n = 2$.

Then we have:

\(\ds 2^2\) \(=\) \(\ds 4\)
\(\ds a\) \(=\) \(\ds 3 \times 2^2 - 1\)
\(\ds \) \(=\) \(\ds 11\)
\(\ds b\) \(=\) \(\ds 3 \times 2^{2 - 1} - 1\)
\(\ds \) \(=\) \(\ds 5\)
\(\ds c\) \(=\) \(\ds 9 \times 2^{2 \times 2 - 1} - 1\)
\(\ds \) \(=\) \(\ds 9 \times 8 - 1\)
\(\ds \) \(=\) \(\ds 71\)

Each of $a, b, c$ are prime.


Thus:

\(\ds 2^2 a b\) \(=\) \(\ds 2^2 \times 11 \times 5\)
\(\ds \) \(=\) \(\ds 220\)
\(\ds 2^2 c\) \(=\) \(\ds 2^2 \times 71\)
\(\ds \) \(=\) \(\ds 284\)

and so $220$ and $284$ form a Thabit pair.

$\blacksquare$