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

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## 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.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $220$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $220$