On amicable pairs
Authors:
Graeme L. Cohen and Herman J. J. te Riele
Journal:
Math. Comp. 67 (1998), 399411
MSC (1991):
Primary 11A25, 11Y70
MathSciNet review:
1458219
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Abstract: Let denote Euler's totient function, i.e., the number of positive integers and prime to . We study pairs of positive integers with such that for some integer . We call these numbers amicable pairs with multiplier , analogously to Carmichael's multiply amicable pairs for the function (which sums all the divisors of ). We have computed all the amicable pairs with larger member and found pairs for which the greatest common divisor is squarefree. With any such pair infinitely many other amicable pairs can be associated. Among these pairs there are socalled primitive amicable pairs. We present a table of the primitive amicable pairs for which the larger member does not exceed . Next, amicable pairs with a given prime structure are studied. It is proved that a relatively prime amicable pair has at least twelve distinct prime factors and that, with the exception of the pair , if one member of a amicable pair has two distinct prime factors, then the other has at least four distinct prime factors. Finally, analogies with construction methods for the classical amicable numbers are shown; application of these methods yields another 79 primitive amicable pairs with larger member , the largest pair consisting of two 46digit numbers.
 1.
Walter
Borho, Eine Schranke für befreundete Zahlen mit gegebener
Teileranzahl, Math. Nachr. 63 (1974), 297–301
(German). MR
0364071 (51 #326)
 2.
W.
Borho, Some large primes and amicable
numbers, Math. Comp. 36
(1981), no. 153, 303–304. MR 595068
(82d:10021), http://dx.doi.org/10.1090/S00255718198105950682
 3.
Sonja
Brentjes and Jan
P. Hogendijk, Notes on Thābit ibn Qurra and his rule for
amicable numbers, Historia Math. 16 (1989),
no. 4, 373–378 (English, with French and German summaries). MR 1040183
(91m:01004), http://dx.doi.org/10.1016/03150860(89)900840
 4.
R. D. Carmichael, Review of History of the Theory of Numbers, Amer. Math. Monthly 26 (1919), 396403.
 5.
G. L. Cohen and H. J. J. te Riele, On amicable pairs [??](with appendix[??]), Research Report R959 (December 1995), School of Mathematical Sciences, University of Technology, Sydney, and CWIReport NMR9524 (November 1995), CWI Amsterdam, ftp://ftp.cwi.nl/pub/CWIreports/NW/NMR9524.ps.Z .
 6.
Richard
K. Guy, Unsolved problems in number theory, 2nd ed., Problem
Books in Mathematics, SpringerVerlag, New York, 1994. Unsolved Problems in
Intuitive Mathematics, I. MR 1299330
(96e:11002)
 7.
Miriam
Hausman, The solution of a special arithmetic equation, Canad.
Math. Bull. 25 (1982), no. 1, 114–117. MR 657659
(83i:10019), http://dx.doi.org/10.4153/CMB1982015x
 8.
T. E. Mason, On amicable numbers and their generalizations, Amer. Math. Monthly 28 (1921), 195200.
 9.
H.
Jager (ed.), Number theory, Noordwijkerhout 1983, Lecture
Notes in Mathematics, vol. 1068, SpringerVerlag, Berlin, 1984. MR 756078
(85i:11001)
 10.
H.
J. J. te Riele, Computation of all the amicable pairs
below 10¹⁰, Math. Comp.
47 (1986), no. 175, 361–368, S9–S40. With a
supplement. MR
842142 (87i:11014), http://dx.doi.org/10.1090/S00255718198608421423
 1.
 W. Borho, Eine Schranke für befreundete Zahlen mit gegebener Teileranzahl, Math. Nachr. 63 (1974), 297301. MR 51:326
 2.
 W. Borho, Some large primes and amicable numbers, Math. Comp. 36 (1981), 303304. MR 82d:10021
 3.
 Sonja Brentjes and Jan P. Hogendijk, Notes on Th\={a}bit ibn Qurra and his rule for amicable numbers, Historia Math. 16 (1989), 373378. MR 91m:01004
 4.
 R. D. Carmichael, Review of History of the Theory of Numbers, Amer. Math. Monthly 26 (1919), 396403.
 5.
 G. L. Cohen and H. J. J. te Riele, On amicable pairs [??](with appendix[??]), Research Report R959 (December 1995), School of Mathematical Sciences, University of Technology, Sydney, and CWIReport NMR9524 (November 1995), CWI Amsterdam, ftp://ftp.cwi.nl/pub/CWIreports/NW/NMR9524.ps.Z .
 6.
 Richard K. Guy, Unsolved Problems in Number Theory, SpringerVerlag, New York, etc., 1994, second edition. MR 96e:11002
 7.
 Miriam Hausman, The solution of a special arithmetic equation, Canad. Math. Bull. 25 (1982), 114117. MR 83i:10019
 8.
 T. E. Mason, On amicable numbers and their generalizations, Amer. Math. Monthly 28 (1921), 195200.
 9.
 H. J. J. te Riele, New very large amicable pairs, Number Theory Noordwijkerhout 1983 (H. Jager, ed.), SpringerVerlag, 1984, pp. 210215. MR 85i:11001
 10.
 H. J. J. te Riele, Computation of all amicable pairs below , Math. Comp. 47 (1986), 361368, S9S40. MR 87i:11014
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Additional Information
Graeme L. Cohen
Affiliation:
School of Mathematical Sciences, University of Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia
Email:
glc@maths.uts.edu.au
Herman J. J. te Riele
Affiliation:
CWI, Department of Modeling, Analysis and Simulation, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands
Email:
herman@cwi.nl
DOI:
http://dx.doi.org/10.1090/S0025571898009338
PII:
S 00255718(98)009338
Keywords:
Euler's totient function,
$\phi $amicable pairs
Received by editor(s):
November 28, 1995
Received by editor(s) in revised form:
May 10, 1996
Article copyright:
© Copyright 1998 American Mathematical Society
