Definition:Thabit Pair

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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:

\(\displaystyle a\) \(=\) \(\displaystyle 3 \times 2^n - 1\)
\(\displaystyle b\) \(=\) \(\displaystyle 3 \times 2^{n - 1} - 1\)
\(\displaystyle c\) \(=\) \(\displaystyle 9 \times 2^{2 n - 1} - 1\)

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.


Also see


Source of Name

This entry was named for Thabit ibn Qurra.


Sources