# 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.

$\blacksquare$