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:

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

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


Thus:

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

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

$\blacksquare$