Thâbit ibn Kurrah's rules is a beautiful result of Thâbit ibn Kurrah dating back to the tenth century (Woepcke 1852; Escott 1946; Dickson 2005, pp. 5 and 39; Borho 1972). Take and suppose that
(1)
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(2)
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(3)
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are all prime. Then are an amicable pair, where is sometimes called a Thâbit ibn Kurrah number. This form was rediscovered by Fermat in 1636 and Descartes in 1638 and generalized by Euler to Euler's rule (Borho 1972).
In order for such numbers to exist, there must be prime for two consecutive , leaving only the possibilities 1, 2, 3, 4, and 6, 7. Of these, is prime for , 4, and 7, giving the amicable pairs (220, 284), (17296, 18416), and (9363584, 9437056).
In fact, various rules can be found that are analogous to Thâbit ibn Kurrah's. Denote a "Thâbit rule" by for given natural numbers and , a prime not dividing , , and polynomials . Then a necessary condition for the set of amicable pairs of the form (, 2) with , prime and a natural number to be infinite is that
(4)
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where is the divisor function (Borho 1972). As a result, (, 2) form an amicable pair, if for some , both
(5)
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for , 2 are prime integers not dividing (Borho 1972).
The following table summarizes some of the known Thâbit ibn Kurrah rules (Borho 1972, te Riele 1974).
72 | 127 | ||
108 | 193 | ||
240 | 449 | ||
252 | 457 | ||
1164 | 2129 | ||
2700 | 5281 | ||
5868 | 10753 | ||
7104 | 13313 | ||
7308 | 14081 | ||
7308 | 14401 | ||
17100 | 33601 | ||
31752 | 57457 | ||
67500 | 134401 | ||
67500 | 134401 | ||
162288 | 311041 | ||
477900 | 950401 | ||
1512300 | 3021761 | ||
6750828 | 13478401 | ||
8436960 | 16329601 | ||
8520192 | 17007103 | ||
18366768 | 36514801 | ||
1199936448 | 2399587741 |