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:
| \(\ds a\) | \(=\) | \(\ds 3 \times 2^n - 1\) | ||||||||||||
| \(\ds b\) | \(=\) | \(\ds 3 \times 2^{n - 1} - 1\) | ||||||||||||
| \(\ds c\) | \(=\) | \(\ds 9 \times 2^{2 n - 1} - 1\) |
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.
Also see
- Results about Thabit pairs can be found here.
Source of Name
This entry was named for Thabit ibn Qurra.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $220$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $220$