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

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