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Thābit ibn Qurra Rule

Thābit ibn Qurra's rule is a beautiful result of Thābit ibn Qurra dating back to the ninth century (Woepcke 1852, Escott 1946, Borho 1972; Dickson 2005, pp. 5 and 39). Take n>=2 and suppose that

h=3·2^n-1
(1)
t=3·2^(n-1)-1
(2)
s=9·2^(2n-1)-1
(3)

are all prime. Then (2^nht,2^ns) are an amicable pair, where h 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).

In order for such numbers to exist, there must be prime 3·2^n-1 for two consecutive n, leaving only the possibilities 1, 2, 3, 4, and 6, 7. Of these, s is prime for n=2, 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 Thābit ibn Qurra's. Denote a "Thābit rule" by T(b_1,b_2,p,F_1,F_2) for given natural numbers b_1 and b_2, a prime p not dividing b_1, b_2, and polynomials F_1(X),F_2(X) in Z[X]. Then a necessary condition for the set of amicable pairs (m_1,m_2) of the form m_i=p^nb_iq_i (i=1, 2) with q_1, q_2 prime and n a natural number to be infinite is that

p/(p-1)=(b_1)/(sigma(b_1))+(b_2)/(sigma(b_2)),
(4)

where sigma(n) is the divisor function (Borho 1972). As a result, m_i=p^nb_iq_i (i=1, 2) form an amicable pair, if for some n>=1, both

q_i=(p^n(p-1)(b_1+b_2))/(sigma(b_i))-1
(5)

for i=1, 2 are prime integers not dividing b_ip (Borho 1972).

The following table summarizes some of the known Thābit ibn Qurra rules T(au,p,(u+1)X,(u+1)sigma(u)X-1) (Borho 1972, te Riele 1974).

ausigma(u)p
2^25·1172127
3^2·7·135·17108193
3^2·5·1311·19240449
3^2·7^2·135·41252457
3^2·7^2·13·195·19311642129
3^4·5·1129·8927005281
3^2·7·13·41·1635·977586810753
3^2·5·19·377·887710413313
3^4·7·11·2913·521730814081
3^2·7^2·13·19·2941·173730814401
3^2·5·13·1929·5691710033601
3^2·7^2·135·53·973175257457
3^2·5^2·13·31149·44967500134401
3^3·5^3·13149·44967500134401
2·7^2·19·2311·13523162288311041
3^4·5·11·5989·5309477900950401
3^4·5·11^2·71709·212915123003021761
3^2·7^2·11·19·43·89293·22961675082813478401
2^3·3117·107·4339843696016329601
2^8257·33023852019217007103
2^3·19·13783·2186511836676836514801
2^7·2634271·28088311999364482399587741

See also

Amicable Pair, Euler's Rule, Riesel Number

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References

Borho, W. "On Thabit ibn Kurrah's Formula for Amicable Numbers." Math. Comput. 26, 571-578, 1972.Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, 2005.Escott, E. B. E. "Amicable Numbers." Scripta Math. 12, 61-72, 1946.Riesel, H. "Lucasian Criteria for the Primality of N=h(2^n)-1." Math. Comput. 23, 869-875, 1969.Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Basel, Switzerland: Birkhäuser, p. 394, 1994.Sloane, N. J. A. Sequence A002235/M0545 in "The On-Line Encyclopedia of Integer Sequences."te Riele, H. J. J. "Four Large Amicable Pairs." Math. Comput. 28, 309-312, 1974.Woepcke, F. J. Asiatique 20, 320-429, 1852.

Cite this as:

Weisstein, Eric W. "Thābit ibn Qurra Rule." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ThabitibnQurraRule.html

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