Computation of All the Amicable Pairs Below 1010 · Computation of All the Amicable Pairs Below 1010 By H. J. J. te Riele Abstract. An efficient exhaustive numerical search method
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MATHEMATICS OF COMPUTATIONVOLUME 47. NUMBER 175JULY I4K6. PACIES .161-36«
Computation of All the Amicable Pairs Below 1010
By H. J. J. te Riele
Abstract. An efficient exhaustive numerical search method for amicable pairs is.described.
With the aid of this method all 1427 amicable pairs with smaller member below 1010 have
been computed, more than 800 pairs being new. This extends previous exhaustive work below
108 by H. Cohen. In three appendices (contained in the supplements section of this issue),
various statistics are given, including an ordered list of all the gcd's of the 1427 amicable pairs
below 1010 (which may be useful in further amicable pair research). Suggested by the
numerical results, a theorem of Borho and Hoffmann for constructing APs has been extended.
1. Introduction. Let a(m) denote the sum of all the divisors of m, including 1 and
m. An amicable pair (AP) is a pair of positive integers (m,n), m < n, such that
a(m) = a(n) = m + n. We note that m is abundant (since a(m) > 2m) and that n
is deficient (since a(n) < 2n). The smallest AP is
(220,284) = (225.11,2271).In order to check whether or not a given positive integer m is the smaller member
of an amicable pair, it seems necessary, at first sight, to compute a(m) and
n := a(m) — m, to check whether n > m (i.e., whether m is abundant), and, if so, to
compute a(n) and compare a(m) with a(n). This involves one or two complete
factorizations, in case m is deficient or abundant, respectively. However, a closer
look reveals that it is often possible to find out whether a given number m is
deficient (hence cannot be the smaller member of an AP) without the need to
factorize it completely. Moreover, once o(m) and n (= a(m) - m) have been
computed, it is often possible to discover that a(n) ¥= a(m) without the need to
factorize n completely.
These considerations have guided the design of an efficient exhaustive numerical
AP search algorithm, the details of which are given in Section 2. With the aid of this
algorithm we have extended Cohen's exhaustive list of all 236 APs with smaller
member below 108 [4] to all 1427 APs with smaller member below 1010. Of these,
601 have been published earlier [6], [7]. The other 826 seem to be new, and are
published here for the first time (9 of them have been communicated to the author
already in 1983 and 1984 by Woods (2), Borho (2) and Lee (5)). Section 3 presents
details of the computations together with several tables collected from this search.
Moreover, a result of Borho and Hoffmann for constructing APs is extended, as was
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SUPPLEMENT S21
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SUPPLEMENT S24
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