Ivars Peterson's MathTrek

February 2, 2004

Amicable Pairs, Divisors, and a New Record

The Pythagoreans of ancient Greece were fascinated by whole numbers. One particular interest involved what we now call amicable numbers.

Amicable numbers come in pairs in which each number is the sum of the proper divisors of the other. The smallest such pair is 220 and 284. The number 220 is evenly divisible by 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, which add up to 284; and 284 is evenly divisible by 1, 2, 4, 71, and 142, which add up to 220. The Pythagorean brotherhood regarded 220 and 284 as numerical symbols of friendship.

The first few amicable pairs are (220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595), (17296, 18416), and (63020, 76084). There are 5,001 amicable pairs in which each number is less than 3.06 x 10¹¹. Mathematicians have conjectured that there are infinitely many amicable pairs.

It's also possible to see what happens for alternative definitions of divisibility. Instead of considering proper divisors (the divisors excluding the number itself but including 1), you can consider so-called unitary divisors. For example, the divisors of 12 are 1, 2, 3, 4, 6, and 12. The unitary divisors (those that have a greatest common divisor of 1) are 1, 3, 4, and 12.

The first unitary amicable pair is 114 and 126. The divisors of 114 are 1, 2, 3, 6, 19, 38, 57, and 114, and its unitary divisors are also 1, 2, 3, 6, 19, 38, 57, and 114. The divisors of 126 are 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, and 126, and its unitary divisors are 1, 2, 7, 9, 14, 18, 63, and 126. Both sets of unitary divisors sum to 240.

The first few unitary amicable pairs are (1140, 1260), (18018, 22302), and (32130, 40446).

Yasutoshi Kohmoto has now set the record for the largest known unitary amicable pair, each member of which has 317 digits:

2⁴ * 3 * 5⁴ * 7⁵ * 11 * 13 * 19² * 23 * 29 * 31 * 97 * 101 * 127 * 137 * 151 * 181 * 191 * 227 * 251 * 313² * 1523 * 17569 * 18119 * 22193 * 42767 * 133157 * 1594471 * 3592427 * 12755767 * 16563721580414291 * 3692133344284919899954037 * 1107640003285475969986211099 * 509326829322602570550995760607650943756709374175283588981851230229946163884862251 * 10645931718799413777310016871802685274740167876616
0269617017644971540715800242987

and

2⁴ * 3 * 5⁴ * 7⁵ * 11 * 13 * 19² * 23 * 29 * 31 * 97 * 101 * 127 * 137 * 151 * 181 * 191 * 227 * 251 * 313² * 1523 * 17569 * 18119 * 22193 * 42767 * 133157 * 1594471 * 3592427 * 12755767 * 16563721580414291 * 3692133344284919899954037 * 1107640003285475969986211099 * 509326829322602570550995760607650943 * 53 * 14013136558801547944108356115369373207909921 * 10645931718799413777310016871802685424680729055
7925899636611749316063659704331051

An inventory of known unitary amicable pairs can be found at http://amicable.homepage.dk/knwnunap.htm. It lists more than 1 million pairs. That's a lot of friendship!

References:

Yasutoshi Kohmoto has a Web page at http://boat.zero.ad.jp/~zbi74583/another02.htm.

Peterson, I. 2001. Appealing numbers. MAA Online (Feb. 26). For an introduction to unitary amicable pairs, see http://mathworld.wolfram.com/UnitaryAmicablePair.html.

Comments are welcome. Please send messages to Ivars Peterson at ip@sciserv.org.

A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the Mathematical Association of America (MAA) book Mathematical Treks: From Surreal Numbers to Magic Circles. See http://www.maa.org/pubs/books/mtr.html.