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 René Descartes, who also rediscovered Thabit's Rule.


Sources