Thabit Pair/Examples/17,296-18,416
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Example of Thabit Pair
$17 \,296$ and $18 \, 416$ form a Thabit pair.
Proof
Let $n = 4$.
Then we have:
\(\ds 2^4\) | \(=\) | \(\ds 16\) | ||||||||||||
\(\ds a\) | \(=\) | \(\ds 3 \times 2^4 - 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 47\) | ||||||||||||
\(\ds b\) | \(=\) | \(\ds 3 \times 2^{4 - 1} - 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 23\) | ||||||||||||
\(\ds c\) | \(=\) | \(\ds 9 \times 2^{2 \times 4 - 1} - 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 9 \times 128 - 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1151\) |
Each of $a, b, c$ are prime.
Thus:
\(\ds 2^4 a b\) | \(=\) | \(\ds 2^4 \times 47 \times 23\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 17 \, 296\) | ||||||||||||
\(\ds 2^4 c\) | \(=\) | \(\ds 2^4 \times 1151\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 18 \, 416\) |
Hence by Thabit's Rule, $17 \, 296$ and $18 \, 416$ form a Thabit pair.
$\blacksquare$
Historical Note
Most of the literature on the subject states that the Thabit pair $\tuple {17 \,296, 18 \, 416}$ was discovered by Ibn al-Banna' al-Marrakushi.
However, as it is generated by Thabit's Rule for the accessibly low number $n = 4$, it is implausible to suppose that it had not in fact been discovered previously by Thabit ibn Qurra himself.
Hence it would be more accurate to say that Ibn al-Banna' rediscovered' it.
In $1636$ it was again rediscovered, this time by Pierre de Fermat, 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): $220$