Chapter II

Review of Related Literature and Studies

The researcher’s topic tackles about the Amicable Pair. In this chapter, you will see some of the

related literature and studies about the topic.

Related Literatures:

Proper Divisors

A positive proper divisor is a positive divisor of a number , excluding itself. For

example, 1, 2, and 3 are positive proper divisors of 6, but 6 itself is not. The number of proper

divisors of is therefore given by

where is the divisor function. For , 2, ..., is therefore given by 0, 1, 1, 2, 1, 3, 1, 3,

2, 3, ... (Sloane's A032741). The largest proper divisors of , 3, ... are 1, 1, 2, 1, 3, 1, 4, 3, 5,

1, ... (Sloane's A032742).

The term "proper divisor" is sometimes used to include negative integer divisors of a

number (excluding ). Using this definition, , , , 1, 2, and 3 are the proper divisors of

6, while and 6 are the improper divisors.

To make matters even more confusing, the proper divisor is often defined so that and 1

are also excluded. Using this alternative definition, the proper divisors of 6 would then be , ,

2, and 3, and the improper divisors would be , , 1, and 6.

(Source: Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved

Problem in Mathematics. New York: Penguin, pp. 8-9, 2004. )

Amicable Numbers

Two numbers are said to be amicable (i.e., friendly) if each one of them is equal to the

sum of the proper divisors of the others (i.e., whole numbers less than the given numbers that

divide the given number with no remainder). For example, 220 have proper divisors 1, 2, 4, 5,

10, 11, 20, 22, 44, 55, and 110. The sum of these divisors is 284. The proper divisors of 284 are

1, 2, 4, 71, and 142. Their sum is 220; so 220 and 284 are amicable. This is the smallest pair of

amicable numbers. (Source: "Amicable numbers." The Gale Encyclopedia of Science. 2008.

Encyclopedia.com. (October 17, 2011).

http://www.encyclopedia.com/article-1G2-2830100095/amicable-numbers.html)

History of Amicable Numbers

The discovery of amicable numbers is attributed to the neo-Pythagorean Greek

philosopher Iamblichus of Chalcis (c. AD 250–330), who credited Pythagoras (582–500 BC)

with the original knowledge of their nature. The Pythagoreans believed that amicable numbers,

like all special numbers, had a profound cosmic significance. A biblical reference (a gift of 220

goats from Jacob to Esau, Genesis 23: 14) is thought by some to indicate an earlier knowledge of

amicable numbers.

No pairs of amicable numbers other than 220 and 284 were discovered by European

mathematicians until 1636, when French mathematician Pierre de Fermat (1601–1665) found the

pair 18, 496 and 17, 296. A century later, Swiss mathematician Leonhard Euler (1707–1783)

made an extensive search and found about 60 additional pairs. Surprisingly, however, he

overlooked the smallest pair after 220 and 284, which is 1184 and 1210. It was subsequently

discovered in 1866 by a 16-year-old boy, Nicolo Paganini.

During the medieval period, Arabian mathematicians preserved and developed the

mathematical knowledge of the ancient Greeks. For example, the polymath Thabit ibn Qurra

(836–901) formulated an ingenious rule for generating amicable number pairs: Let a = 3(2 n) – 1,

b = 3(2n-1) – 1, and c = 9(22n-1) – 1; then, if a, b, and c are primes, 2nab and 2nc are amicable. This

rule produces 220 and 284 when n is 2. When n is 3, c is not a prime, and the resulting numbers

are not amicable. For n = 4, it produces Fermat’s pair, 17, 296 and 18, 416, skipping over

Paganini’s pair and others.

Other scientists who have studied amicable numbers throughout history are Spanish

mathematician Al Madshritti (died 1007), Islamic mathematician Abu Mansur Tahir al-Baghdadi

(980–1037), French mathematician and philosopher René Descartes (1596–1650), and Swiss

mathematician Leonhard Euler (1707–1783).

Professionals and amateurs alike have for centuries enjoyed seeking them (Amicable

Pairs) and exploring their properties. (Source: "Amicable numbers." The Gale Encyclopedia of

Science. 2008. Encyclopedia.com. (October 17, 2011). http://www.encyclopedia.com/article-

1G2-2830100095/amicable-numbers.html)

Amicable Pair

An amicable pair is a pair of positive integers (m, n), , such that (m)= (n)=m+n,

where (.) denotes the sum of divisors function. These number pairs have a long and interesting

history (cf. [5]). Euler [14] was the first who systematically studied amicable pairs, and a great

part of the known pairs were found with his methods and the use of electronic computers.

(Source: Herman J. J. Te Riele.January 1984. Mathematics of Competition, Volume 42 pages

219-223, “On Generating New Amicable Pairs from given Amicable Pairs”)

An amicable pair consists of two integers 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 (Hoffman 1998, p. 45), although this nomenclature is to be

discouraged since the numbers more commonly known as friendly pairs are defined by a

different, albeit related, criterion. Symbolically, amicable pairs satisfy

where

is the restricted divisor function. Equivalently, an amicable pair satisfies

where is the divisor function. The smallest amicable pair is (220, 284) which has

factorizations

giving restricted divisor functions

The quantity

in this case, , is called the pair sum.

Rules for producing amicable pairs include the Thâbit ibn Kurrah rule rediscovered by

Fermat and Descartes and extended by Euler to Euler's rule. A further extension not previously

noticed was discovered by Borho (1972).

Pomerance (1981) has proved that

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

Let an amicable pair be denoted , and take . is called a regular amicable pair of

type if

where is the greatest common divisor,

and are squarefree, then the number of prime factors of and are and . Pairs which are

not regular are called irregular or exotic (te Riele 1986). There are no regular pairs of type

for . If and

is even, then cannot be an amicable pair (Lee 1969). The minimal and maximal values of

found by te Riele (1986) were

and

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

te Riele (1986) found no amicable -tuples having the same pair sum for . 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.

The earliest known odd amicable numbers all were divisible by 3. This led Bratley and

McKay (1968) to conjecture that there are no amicable pairs coprime to 6 (Guy 1994, p.  56).

However, Battiato and Borho (1988) found a counterexample, and now many amicable pairs are

known which are not divisible by 6 (Pedersen). The smallest known example of this kind is the

amicable pair (42262694537514864075544955198125,

42405817271188606697466971841875), each number of which has 32 digits.

A search was then begun for amicable pairs coprime to 30. The first example was found

by Y. Kohmoto in 1997, consisting of a pair of numbers each having 193 digits (Pedersen).

Kohmoto subsequently found two other examples, and te Riele and Pedersen used two of

Kohmoto's examples to calculated 243 type- pairs coprime to 30 by means of a method

which generates type- pairs from a type- pairs.

No amicable pairs which are coprime to are currently known.

The following table summarizes the largest known amicable pairs discovered in recent years.

The largest of these is obtained by defining

then , , and are all primes, and the numbers

are an amicable pair, with each member having decimal digits (Jobling 2005).

digits date reference

4829 Oct. 4, 1997 M. García

8684 Jun. 6, 2003 Jobling and Walker 2003

16563 May 12, 2004 Walker et al. 2004

17326 May 12, 2004 Walker et al. 2004

24073 Mar. 10, 2005 Jobling 2005

Amicable pairs in Gaussian integers also exist, for example

and

(Source: (Weisstein, Eric W. “Amicable Pair.” From MathWorld—A Wolfram Web Resource.

http://mathworld.wolfram.com/AmicablePair.html)

Related Studies:

Friendly Number

In number theory, friendly numbers are numbers in friendly pair and are two or more

natural numbers with a common abundancy, the ratio between the sum of divisors of a number

and the number itself. Two numbers with the same abundancy form a friendly pair; n numbers

with the same abundancy form a friendly n-tuple.

Being mutually friendly is an equivalence relation, and thus induces a partition of the

positive naturals into clubs (equivalence classes) of mutually friendly numbers.

A number that is not part of any friendly pair is called solitary.

There are some numbers that can easily be proved to be solitary, but the status of

numbers 10, 14, 15, 20, and many others remains unknown .The numbers known to be friendly

are given by 6, 12, 24, 28, 30, 40, 42, 56, 60, ... (Sloane's A074902). (Source: Hickerson.2002.

http://mathworld.wolfram.com/FriendlyNumber.html)

Sociable Numbers

Sociable numbers are generalizations of the concepts of amicable numbers and perfect

numbers. A set of sociable numbers is a kind of aliquot sequence, or a sequence of numbers each

of whose numbers is the sum of the factors of the preceding number, excluding the preceding

number itself. For the sequence to be sociable, the sequence must be cyclic, eventually returning

to its starting point.

The period of the sequence, or order of the set of sociable numbers, is the number of

numbers in this cycle.

If the period of the sequence is 1, the number is a sociable number of order 1, or a perfect

number—for example, the proper divisors of 6 are 1, 2, and 3, whose sum is again 6. A pair of

amicable numbers is a set of sociable numbers of order 2. There are no known sociable numbers

of order 3.

It is an open question whether all numbers end up at either a sociable number or at a

prime (and hence 1). Or equivalently, whether there exists a number whose aliquot sequence

never terminates.

(Source: H. Cohen, On amicable and sociable numbers, Math. Comp. 24 (1970), pp. 423-429)

Happy Numbers

A happy number is defined by the following process. Starting with any positive integer,

replace the number by the sum of the squares of its digits, and repeat the process until the

number equals 1 (where it will stay), or it loops endlessly in a cycle which does not include 1.

Those numbers for which this process ends in 1 are happy numbers, while those that do not end

in 1 are unhappy numbers (or sad numbers).

(Source: Guy, Richard (2004). Unsolved Problems in Number Theory (3rd ed.). )

Unitary Amicable Pair

A pair of numbers and such that

where is the unitary divisor function. Hagis (1971) and García (1987) give 82 such pairs.

The first few are (114, 126), (1140, 1260), (18018, 22302), (32130, 40446), ... (Sloane's

A002952 and A002953; Pedersen).

On Jan. 30, 2004, Y. Kohmoto discovered the largest known unitary amicable pair, where each

member has 317 digits.

Kohmoto calls a unitary amicable pair whose members are squareful a proper unitary amicable

pair.

(Source: Pedersen, J. M. "Known Unitary Amicable Pairs."

http://amicable.homepage.dk/knwnunap.htm.)

"Amicable Pair: An Investigatory"

Review of Related Literature and Studies

Presented to:

Ma’am Maribel Santillana Siao

Faculty and Staff ofMindanao State University

Institute of Science EducationScience High School

In Partial fulfillmentof course requirement

in Research

Gandarosa, Norhanifa P.

MARCH 2012