AMICABLE NUMBERS |
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Introduction
Pythagoras when asked, "What is a friend", replied that a friend is one "who is the other I" such as 220 and 284. The numbers 220 and 284 form the smallest pair of amicable numbers (also known as friendly numbers) known to Pythagoras.
Two numbers are called Amicable (or friendly) if each equals to the sum of the aliquot divisors of the other (aliquot divisors means all the divisors excluding the number itself). For example aliquot divisors of number 220 are 1,2,4,5,10,11,20,22,44,55 and 110. The aliquot divisors of number 284 are 1,2,4,71 and 142.
If we represent an amicable pair (AP) by (m, n,) and sum of aliquot divisors of m and n by s (m) and s (n) respectively, then for amicable pair (220, 284) we get
s (m) = s (220) = 1+2+4+5+10+11+20+22+44+55+110
= 284 = n
s (n) = s (284) = 1+2+4+71+142
= 220 = m
It can be seen that each amicable number has the power to generate another, thus symbolizing mutual harmony, perfect friendship and love. The smallest amicable pair (220, 284) is known from antiquity and so much significance was attached to it that the possessor of one was assured of close friendship with the possessor of the other number of the pair and so much so some marriages have been made in the past on the basis of amicable numbers.
It was not until 1636 that the great Pierre de Fermat discovered another pair of amicable numbers (17296, 18416). Later Descartes gave the third pair of amicable numbers i.e. (9363584, 9437056). These results were actually rediscoveries of numbers known to Arab mathematicians. In the 18^{th} century great Euler drew up a list of 64 amiable pairs (two of which later shown to be unfriendly). B.N.I. Paganini, a 16 years Old Italian, startled the mathematical world in 1866 by announcing that the numbers 1184 and 1210 were friendly. It was the second lowest pair and had been completely overlooked until then, Even Eulers list of Amicable pairs does not contain it. Today(12 April, 2021) about 1226909879 pairs of amicable numbers is known.
There are many rules for finding amicable numbers sets in contrast with the single formula for perfect numbers. One simple rule given by Arab Thabit ibn Korrah in 9^{th} century is given below:
Take any power of 2, such as 2^{n }where n>1 and form the numbers
h = 3.2^{n } -1
t = 3.2^{n-1 } -1
s = 9.2^{2n-1 } -1
If h, t and s are all primes then 2^{h }ht and 2^{n }s are amicable.
For example when n=2, we get h=11, t=5 and s=71, which are all primes hence 2^{n} h t = 220 and 2^{n }s = 284 are amicable numbers. Similarly when n = 4, we get the amicable pair (17296, 18416).
Distribution of Amicable Numbers
All amicable number pairs below 10^{10 }have been compiled and published by H.J.J. te. Riele [1]. There are 1427 amicable pairs below 10^{10 }. Subsequently all Amicable Pairs with the smallest member below 10^{20} have been found. The details of all known Amicable Pairs can be found here. The distribution of Amicable Pairs up to 10^{20 }is given in table 1.
Table 1
Distribution of Amicable pairs below 10^{20 }.
X |
No. of Amicable Pairs whose smaller number is less than X |
10^{3} |
1 |
10^{4} |
5 |
10^{5} |
13 |
10^{6} |
42 |
10^{7} |
108 |
10^{8} |
236 |
10^{9} |
586 |
10^{10} |
1427 |
10^{11} |
3340 |
10^{12} |
7642 |
10^{13} |
17519 |
10^{14} |
39374 |
10^{15} |
87102 |
10^{16} |
190775 |
10^{17} |
415523 |
10^{18} |
901312 |
10^{19} |
1947667 |
10^{20} |
4197267 |
Looking to distribution of Amicable pairs up to 10^{10} , it was observed [4] that if the number of amicable pairs (APs) whose smaller number is less than 10^{n }is y, then the number of amicable pairs whose smaller number is less than 10^{n+1 }shall be at least 2y, where n is any positive integer number greater than 2. For example, number of amicable pairs below 10^{9 }is 586, so minimum number of amicable pairs below 10^{9+1 }= 10^{10 }must be 2*586 = 1172 which is correct as it less than the actual number of amicable pairs below 10^{10 }i.e. 1427.
It is now seen that observation made in [4] still holds good for number of amicable pairs up to 10^{20} . Based on this observation, number of amicable pairs up to 10^{21} must be at least 2*4197267 = 8394534. It can be seen here that the total number of amicable pairs already known up to 10^{21} are 7167071.
Some fascinating Observations
Some of the other fascinating observations regarding amicable pairs are given below.
(1) There is no amicable pair in which one of the two numbers is a square.
(2) There are some amicable pairs (m, n), in which the sum of digits of m and n is equal [4]. For example, consider amicable pair ( 69615, 87633),
Sum of digits of 69615 = 6+9+6+1+5 = 27
Sum of digits of 87633 = 8+7+6+3+3 = 27
Other examples of such amicable pairs are (100485, 124155), (1358595, 1486845) etc. Number enthusiasts can find more such amicable pairs. There are 427 such Amicable Pairs in first 5000 Amicable Pairs. The list of these 427 Amicable pairs is available here.
(3) There are some amicable pairs (m, n) in which both m and n are divisible by the sum of their digits [4].
For example, consider amicable pairs (2620, 2924), where 2620 is divisible by 2+6+2+0 = 10 (i.e. 2620/10 = 262) and 2924 is divisible by 2+9+2+4 = 17 (i.e. 2924/17 = 172). The details of such pairs are given below under the heading Harshad Amicable Pairs.
(4) The minimal and maximal value of m/n for an amicable pair (m, n) where m<n is given in [1] as 0.6979 and 0.999858 , which corresponds to amicable pair #567 (938304290, 1344480478) and amicable pair #1010 (4000783984, 4001351168).
The minimal and maximal values of m/n based on all known amicable pairs up to Dec 2001 (2494343) as conveyed by Jan Munch Pedersen (email dated 05-01-2002) are 0.598343 and 0.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999998.
#13 (79750, 88730)
#695(1558818261, 1596205611)
#605(106930732, 1142071892)
#501 (664747083, 673747893)
#22(196724, 202444)
#7 (12285, 14595)
#8 (17296, 18416)
#4909(290142314847, 292821792417)
#28(469028, 486178)
#2902(68606181189, 70516785339)
Note that in the Amicable Pair say for k = 7 i.e. #4909 (290142314847, 292821792417) , #4909 represent Rank number and the amicable numbers 290142314847 and 292821792417ends in k i.e. 7 and this is the smallest pair in which both numbers in the pair ends in 7.
Harshad Amicable Pairs
Harshad (or Niven ) numbers are those numbers which are divisible by their sum of the digits. For example 1729 ( 19*91) is divisible by 1+7+2+9 =19, so 1729 is a Harshad number.
We define Harshad Amicable Pair as an Amicable Pair (m, n) , such that both m and n are Harshad numbers. For example, consider amicable pair (2620, 2924), where 2620 is divisible by 2+6+2+0 = 10 (i.e. 2620/10 = 262) and 2924 is divisible by 2+9+2+4 = 17 (i.e. 2924/17 = 172). So both 2620 and 2924 are Harshad numbers and hence the Amicable Pair (2620, 2924) is Harshad Amicable Pair. Other examples are (10634085,14084763), (23389695, 25132545), (34256222, 35997346) etc. The search for such amicable pairs can be a good past time. There are 192 Harshad Amicable Pairs in first 5000 Amicable Pairs. The list of these 192 Harshad Amicable pairs is available here.
Happy Amicable Pairs
If you iterate the process of summing the squares of the decimal digits of a number and if the process terminates in 1, then the original number is called a Happy number. For example 7 -> 49 -> 97 -> 130 -> 10 -> 1.
We define Happy Amicable Pair as an Amicable Pair (m, n) , such that both m and n are Happy numbers. For example, consider amicable pair (10572550, 10854650), where 10572550 -> 129 -> 86 -> 100 -> 1 and 10854650 -> 167 -> 86 -> 100 -> 1. So both 10572550 and 10854650 are Happy numbers and hence the amicable pair (10572550, 10854650) is Happy Amicable Pair. Other examples are (32685250, 34538270), (35361326, 40117714), (35390008, 39259592) etc. The search for such amicable pairs can be a good past time. There are 111 Happy Amicable Pairs in first 5000 Amicable Pairs. The list of these 111 Happy Amicable Pairs is available here.
References for further reading
(1) H.J.J. te . Riele, "computations of all the available pairs below 10^{10 }, Math Comp. 47 (1986). pp. 361 – 368 and supplement S9 – S40.
(2) H.J.J. te. Riele, w. Borho, S.Batliato, H. Haffmann, E.J.Lee, "Table of Amicable Pairs between 10^{10 }and 10^{52" }CWI report Sept., 1986.
(3) L.E.Dickson, "History of the theory of Numbers" Vol. I, U.S.A.
(4) S.S.Gupta, "Amicable Number Pairs and Triplets" , Parardh , Vol. 1, No.1,1988.
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