Amicable Pair

An amicable pair consists of two Integers $m, n$ for which the sum of Proper Divisors (the Divisors excluding the number itself) of one number equals the other. Amicable pairs are occasionally called Friendly Pairs, although this nomenclature is to be discouraged since Friendly Pairs are defined by a different, if related, criterion. Symbolically, amicable pairs satisfy

$$s(m) = n$$ (1)

$$s(n) = m,$$ (2)

where $s(n)$ is the Restricted Divisor Function or, equivalently,

$$\sigma(m)=\sigma(n)=s(m)+s(n)=m+n,$$ (3)

where $\sigma(n)$ is the Divisor Function. The smallest amicable pair is (220, 284) which has factorizations

$$220 = 11 \cdot 5 \cdot 2^2$$ (4)

$$284 = 71 \cdot 2^2$$ (5)

giving Restricted Divisor Functions

$$s(220) = \sum \{1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110\}$$

$$= 284$$ (6)

$$s(284) = \sum \{1, 2, 4, 71, 142\}$$

$$= 220.$$ (7)

The quantity

$$\sigma(m)=\sigma(n)=s(m)+s(n),$$ (8)

in this case, $220+284=504$, is called the Pair Sum.

In 1636, Fermat found the pair (17296, 18416) and in 1638, Descartes found (9363584, 9437056). By 1747, Euler had found 30 pairs, a number which he later extended to 60. There were 390 known as of 1946 (Scott 1946). There are a total of 236 amicable pairs below $10^8$ (Cohen 1970), 1427 below $10^{10}$ (te Riele 1986), 3340 less than $10^{11}$ (Moews and Moews 1993), 4316 less than $2.01\times 10^{11}$ (Moews and Moews), and 5001 less than $\approx 3.06\times 10^{11}$ (Moews and Moews).

The first few amicable pairs are (220, 284), (1184, 1210), (2620, 2924) (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595), (17296, 18416), (63020, 76084), ... (Sloane's A002025 and A002046). An exhaustive tabulation is maintained by D. Moews.

Let an amicable pair be denoted $(m,n)$ with $m<n$. $(m,n)$ is called a regular amicable pair of type $(i,j)$ if

$$(m,n)=(gM,gN),$$ (9)

where $g\equiv\mathop{\rm GCD}\nolimits (m,n)$ is the Greatest Common Divisor,

$$\mathop{\rm GCD}\nolimits (g,M)=\mathop{\rm GCD}\nolimits (g,N)=1,$$ (10)

$M$ and $N$ are Squarefree, then the number of Prime factors of $M$ and $N$ are $i$ and $j$. Pairs which are not regular are called irregular or exotic (te Riele 1986). There are no regular pairs of type $(1,j)$ for $j\geq 1$. If $m\equiv 0\ \left({{\rm mod\ } {6}}\right)$ and

$$n=\sigma(m)-m$$ (11)

is Even, then $(m,n)$ cannot be an amicable pair (Lee 1969). The minimal and maximal values of $m/n$ found by te Riele (1986) were

$$938304290/1344480478=0.697893577\ldots$$ (12)

and

$$4000783984/4001351168=0.9998582518\ldots.$$ (13)

te Riele (1986) also found 37 pairs of amicable pairs having the same Pair Sum. The first such pair is (609928, 686072) and (643336, 652664), which has the Pair Sum

$$\sigma(m)=\sigma(n)=m+n=1{,}296{,}000.$$ (14)

te Riele (1986) found no amicable $n$-tuples having the same Pair Sum for $n>2$. However, Moews and Moews found a triple in 1993, and te Riele found a quadruple in 1995. In November 1997, a quintuple and sextuple were discovered. The sextuple is (1953433861918, 2216492794082), (1968039941816, 2201886714184), (1981957651366, 2187969004634), (1993501042130, 2176425613870), (2046897812505, 2123028843495), (2068113162038, 2101813493962), all having Pair Sum 4169926656000. Amazingly, the sextuple is smaller than any known quadruple or quintuple, and is likely smaller than any quintuple.

On October 4, 1997, Mariano Garcia found the largest known amicable pair, each of whose members has 4829 Digits. The new pair is

$$N_1 = CM[(P+Q)P^{89}-1]$$ (15)

$$N_2 = CQ[(P-M)P^{89}-1],$$ (16)

where

$$C = 2^{11}P^{89}$$ (17)

$$M = 287155430510003638403359267$$ (18)

$$P = 574451143340278962374313859$$ (19)

$$Q = 136272576607912041393307632916794623.$$ (20)

$P$, $Q$, $(P+Q)P^{89}-1$, and $(P-M)P^{89}-1$ are Prime.

Pomerance (1981) has proved that

$$[\hbox{amicable numbers }\leq n]< n e^{-{[\ln(n)]}^{1/3}}$$ (21)

for large enough $n$ (Guy 1994). No nonfinite lower bound has been proven.

See also Amicable Quadruple, Amicable Triple, Augmented Amicable Pair, Breeder, Crowd, Euler's Rule, Friendly Pair, Multiamicable Numbers, Pair Sum, Quasiamicable Pair, Sociable Numbers, Unitary Amicable Pair

References

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