Thabit Pair/Examples/220-284

Example of Thabit Pair

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

Proof

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

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.

$\blacksquare$