Thabit Pair/Examples/17,296-18,416

Example of Thabit Pair

$17 \,296$ and $18 \, 416$ form a Thabit pair.

Proof

Let $n = 4$.

Then we have:

 $\displaystyle 2^4$ $=$ $\displaystyle 16$ $\displaystyle a$ $=$ $\displaystyle 3 \times 2^4 - 1$ $\displaystyle$ $=$ $\displaystyle 47$ $\displaystyle b$ $=$ $\displaystyle 3 \times 2^{4 - 1} - 1$ $\displaystyle$ $=$ $\displaystyle 23$ $\displaystyle c$ $=$ $\displaystyle 9 \times 2^{2 \times 4 - 1} - 1$ $\displaystyle$ $=$ $\displaystyle 9 \times 128 - 1$ $\displaystyle$ $=$ $\displaystyle 1151$

Each of $a, b, c$ are prime.

Thus:

 $\displaystyle 2^4 a b$ $=$ $\displaystyle 2^4 \times 47 \times 23$ $\displaystyle$ $=$ $\displaystyle 17 \, 296$ $\displaystyle 2^4 c$ $=$ $\displaystyle 2^4 \times 1151$ $\displaystyle$ $=$ $\displaystyle 18 \, 416$

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.