Thabit number

An integer of the form $3\cdot 2^n-1$, or $2^{n+1}+2^n-1$. They are listed in A055010 of Sloane’s OEIS. The Thabit numbers are a subset of the Proth numbers.

The mathematician and astronomer Thabit ibn Qurra studied these numbers in search of a formula for amicable pairs. He found that when two consecutive Thabit numbers are also prime numbers (corresponding to indices $n$ and $n-1$) and $9\cdot 2^{2n-1}-1$ is a prime number, too, then these numbers multiplied by $2^n$ will reveal an amicable pair. The only $n$ known to fit these criteria are 2, 4 and 7. The largest Thabit number known to be prime corresponds to index 2312734, its immediate lower neighbor is composite.

It is conjectured that the nimfactorial of a Thabit number is always 2.

Title	Thabit number
Canonical name	ThabitNumber
Date of creation	2013-03-22 15:52:58
Last modified on	2013-03-22 15:52:58
Owner	Mravinci (12996)
Last modified by	Mravinci (12996)
Numerical id	5
Author	Mravinci (12996)
Entry type	Definition
Classification	msc 11A05
Synonym	Thabit ibn Kurra number
Synonym	Thabit ibn Kurrah number
Synonym	Thabit ibn Qurra number
Synonym	Thabit ibn Qurrah number
Synonym	Thabit bin Kurra number
Synonym	Thabit bin Kurrah number
Synonym	Thabit bin Qurra number
Synonym	Thabit bin Qurrah number
Related topic	AFormulaForAmicablePairs