Thabit Pair/Examples/220-284

Example of Thabit Pair

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

Proof

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

$\map {\sigma_1} {220} = \map {\sigma_1} {284} = 504 = 220 + 284$

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.

$\blacksquare$