On -amicable pairs

Authors:
Graeme L. Cohen and Herman J. J. te Riele

Journal:
Math. Comp. **67** (1998), 399-411

MSC (1991):
Primary 11A25, 11Y70

DOI:
https://doi.org/10.1090/S0025-5718-98-00933-8

MathSciNet review:
1458219

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Abstract | References | Similar Articles | Additional Information

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 so-called 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 46-digit numbers.

**1.**W. Borho,*Eine Schranke für befreundete Zahlen mit gegebener Teileranzahl*, Math. Nachr.**63**(1974), 297-301. MR**51:326****2.**W. Borho,*Some large primes and amicable numbers*, Math. Comp.**36**(1981), 303-304. 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), 373-378. MR**91m:01004****4.**R. D. Carmichael,*Review of*, Amer. Math. Monthly*History of the Theory of Numbers***26**(1919), 396-403.**5.**G. L. Cohen and H. J. J. te Riele,*On -amicable pairs [??](with appendix[??])*, Research Report R95-9 (December 1995), School of Mathematical Sciences, University of Technology, Sydney, and CWI-Report NM-R9524 (November 1995), CWI Amsterdam,`ftp://ftp.cwi.nl/pub/CWIreports/NW/NM-R9524.ps.Z`.**6.**Richard K. Guy,*Unsolved Problems in Number Theory*, Springer-Verlag, New York, etc., 1994, second edition. MR**96e:11002****7.**Miriam Hausman,*The solution of a special arithmetic equation*, Canad. Math. Bull.**25**(1982), 114-117. MR**83i:10019****8.**T. E. Mason,*On amicable numbers and their generalizations*, Amer. Math. Monthly**28**(1921), 195-200.**9.**H. J. J. te Riele,*New very large amicable pairs*, Number Theory Noordwijkerhout 1983 (H. Jager, ed.), Springer-Verlag, 1984, pp. 210-215. MR**85i:11001****10.**H. J. J. te Riele,*Computation of all amicable pairs below*, Math. Comp.**47**(1986), 361-368, S9-S40. 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:
https://doi.org/10.1090/S0025-5718-98-00933-8

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