Thābit ibn Qurra Rule

Thābit ibn Qurra's rule gives a sufficient condition for constructing an amicable pair. The rule dates back to the ninth century (Woepcke 1852, Escott 1946, Borho 1972; Dickson 2005, pp. 5 and 39). Take an integer and suppose that

			(1)
			(2)
			(3)

are all prime. Then are an amicable pair, where is sometimes called a Thābit ibn Qurra number. This form was rediscovered by Fermat in 1636 and Descartes in 1638 and generalized by Euler to Euler's rule (Borho 1972).

The rule requires that both and occur among the exponents for which is prime (OEIS A002235). For , 4, and 7, all three numbers , , and are prime, giving the amicable pairs (220, 284), (17296, 18416), and (9363584, 9437056).

In fact, various rules can be found that are analogous to Thābit ibn Qurra's. Denote a "Thābit rule" by for given positive integers and , a prime not dividing , and polynomials , with for , 2, and fixed , independent of . Then a necessary condition for the set of amicable pairs of the form (, 2) with , prime and a positive integer to be infinite is that

(4)

where is the divisor function (Borho 1972). Assuming this balance condition, the numbers (, 2) form an amicable pair if and for some positive integer , both

(5)

for , 2 are prime integers not dividing (Borho 1972).

The following table summarizes some of the known Thābit ibn Qurra 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