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:
\(\ds 2^7\) | \(=\) | \(\ds 128\) | ||||||||||||
\(\ds a\) | \(=\) | \(\ds 3 \times 2^7 - 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 383\) | ||||||||||||
\(\ds b\) | \(=\) | \(\ds 3 \times 2^{7 - 1} - 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 191\) | ||||||||||||
\(\ds c\) | \(=\) | \(\ds 9 \times 2^{2 \times 7 - 1} - 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 9 \times 8192 - 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 73 \, 727\) |
Each of $a, b, c$ are prime.
Thus:
\(\ds 2^7 a b\) | \(=\) | \(\ds 2^7 \times 383 \times 191\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 9 \, 363 \, 584\) | ||||||||||||
\(\ds 2^7 c\) | \(=\) | \(\ds 2^7 \times 73 \, 727\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 9 \, 437 \, 056\) |
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
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $220$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $220$