# Thabit Pair/Examples/9,363,584-9,437,056

## Example of Thabit Pair

$9 \, 363 \, 584$ and $9 \, 437 \, 056$ form a Thabit pair.

## Proof

Let $n = 7$.

Then we have:

 $\displaystyle 2^7$ $=$ $\displaystyle 128$ $\displaystyle a$ $=$ $\displaystyle 3 \times 2^7 - 1$ $\displaystyle$ $=$ $\displaystyle 383$ $\displaystyle b$ $=$ $\displaystyle 3 \times 2^{7 - 1} - 1$ $\displaystyle$ $=$ $\displaystyle 191$ $\displaystyle c$ $=$ $\displaystyle 9 \times 2^{2 \times 7 - 1} - 1$ $\displaystyle$ $=$ $\displaystyle 9 \times 8192 - 1$ $\displaystyle$ $=$ $\displaystyle 73 \, 727$

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

Thus:

 $\displaystyle 2^7 a b$ $=$ $\displaystyle 2^7 \times 383 \times 191$ $\displaystyle$ $=$ $\displaystyle 9 \, 363 \, 584$ $\displaystyle 2^7 c$ $=$ $\displaystyle 2^7 \times 73 \, 727$ $\displaystyle$ $=$ $\displaystyle 9 \, 437 \, 056$

Hence by Thabit's Rule, $9 \, 363 \, 584$ and $9 \, 437 \, 056$ form a Thabit pair.

$\blacksquare$

## Historical Note

The Thabit pair $\tuple {9 \, 363 \, 584, 9 \, 437 \, 056}$ was originally discovered by Thabit ibn Qurra, as a result of applying Thabit's Rule using $n = 7$.

It was rediscovered by RenĂ© Descartes, who also rediscovered Thabit's Rule.