# 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:

\(\displaystyle 2^4\) | \(=\) | \(\displaystyle 16\) | |||||||||||

\(\displaystyle a\) | \(=\) | \(\displaystyle 3 \times 2^4 - 1\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 47\) | |||||||||||

\(\displaystyle b\) | \(=\) | \(\displaystyle 3 \times 2^{4 - 1} - 1\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 23\) | |||||||||||

\(\displaystyle c\) | \(=\) | \(\displaystyle 9 \times 2^{2 \times 4 - 1} - 1\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 9 \times 128 - 1\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 1151\) |

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.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $220$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $220$