Title of the page: Abstract of: A new method for finding amicable pairs
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Abstract of: A new method for finding amicable pairs
NM-R9512
H.J.J. te Riele ;
1995, NM-R9512, ISSN 0169-0388
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Abstract
Let σ(x) denote the sum of all divisors of the (positive) integer x. An amicable pair is a pair of integers (m,n) with m<n such that σ(m)=σ(n)=m+n. The smallest amicable pair is (220,284). A new method for finding amicable pairs is presented, based on the following observation of Erdős: For given s, let x1, x2, … be solutions of the equation σ(x)=s, then any pair (xi,xj) for which xi+xj=s is amicable.
The problem here is to find numbers s for which the equation σ(x)=s has many solutions. From inspection of tables of known amicable pairs and their pair sums one learns that certain smooth numbers s (i.e., numbers with only small prime divisors) are good candidates. With the help of a precomputed table of σ (pe)-values, many solutions of the equation σ(x)=s were found by checking divisibility of s by the tabled σ-values in a recursive way. In the set of solutions found, pairs were traced which sum up to s.
From 1850 smooth numbers s satisfying 4×1011<s<1012 we found 116 new amicable pairs with this algorithm.
After the submission of this paper to the Vancouver Conference Mathematics of Computation 1943--1993, the computations have been extended and yielded many more new amicable pairs. In particular, the first quadruple of amicable pairs with the same pair sum (namely 16!) was found. A list is given of 587 amicable pairs with smaller member between 2.01×1011 and 1012, of which 565 pairs seem to be new.
CWI Project(s):
Computational number theory
Keywords:
Amicable pairs
sum of divisors
H.J.J. te Riele ;
1995, NM-R9512, ISSN 0169-0388
PDF-file
compressed PostScript
( 115 KB )
( 20 pages )
Abstract
Let σ(x) denote the sum of all divisors of the (positive) integer x. An amicable pair is a pair of integers (m,n) with m<n such that σ(m)=σ(n)=m+n. The smallest amicable pair is (220,284). A new method for finding amicable pairs is presented, based on the following observation of Erdős: For given s, let x1, x2, … be solutions of the equation σ(x)=s, then any pair (xi,xj) for which xi+xj=s is amicable.
The problem here is to find numbers s for which the equation σ(x)=s has many solutions. From inspection of tables of known amicable pairs and their pair sums one learns that certain smooth numbers s (i.e., numbers with only small prime divisors) are good candidates. With the help of a precomputed table of σ (pe)-values, many solutions of the equation σ(x)=s were found by checking divisibility of s by the tabled σ-values in a recursive way. In the set of solutions found, pairs were traced which sum up to s.
From 1850 smooth numbers s satisfying 4×1011<s<1012 we found 116 new amicable pairs with this algorithm.
After the submission of this paper to the Vancouver Conference Mathematics of Computation 1943--1993, the computations have been extended and yielded many more new amicable pairs. In particular, the first quadruple of amicable pairs with the same pair sum (namely 16!) was found. A list is given of 587 amicable pairs with smaller member between 2.01×1011 and 1012, of which 565 pairs seem to be new.
CWI Project(s):
Computational number theoryKeywords:
Amicable pairs
sum of divisors