Definition:Thabit Pair

Definition

A Thabit pair is an amicable pair formed by an application of Thabit's Rule:

$\tuple {2^n a b, 2^n c}$

where $n$ be a positive integer such that:

 $\ds a$ $=$ $\ds 3 \times 2^n - 1$ $\ds b$ $=$ $\ds 3 \times 2^{n - 1} - 1$ $\ds c$ $=$ $\ds 9 \times 2^{2 n - 1} - 1$

are all prime.

Examples

$220$ and $284$

$220$ and $284$ form a Thabit pair.

$17 \, 296$ and $18 \, 416$

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

$9 \, 363 \, 584$ and $9 \, 437 \, 056$

$9 \, 363 \, 584$ and $9 \, 437 \, 056$ form a Thabit pair.

These are the only Thabit pairs known.

Source of Name

This entry was named for Thabit ibn Qurra.