Thâbit ibn Kurrah rule

Thâbit ibn Kurrah rule

Thâbit ibn Kurrah rule is a method for discovering amicable numbers invented in the tenth century by the Arab mathematician Thâbit ibn Kurrah. A later generalization of this rule is Euler's rule.

The rule is given in terms of Thâbit ibn Kurrah numbers. For any natural number n, the nth Thâbit ibn Kurrah number is Kn = 3×2n − 1. The first ten Thâbit ibn Kurrah numbers are 2, 5, 11, 23, 47, 95, 191, 383, 767, and 1535.

Thâbit ibn Kurrah showed that if Kn, Kn−1, and 3×K2n−1 + 2 are all prime, then the pair (2n×Kn×Kn−1, 2n×(3×K2n−1 + 2)) is amicable.

The hypothesis is met in only three cases, n = 2, 4, and 7, giving amicable pairs (220, 284), (17296, 18416), and (9363584, 9437056).

Reference:
* [http://mathworld.wolfram.com/ThabitibnKurrahRule.html mathworld]


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