内容概述
两个整数如果各等于对方真因子之和，就称为亲和数。毕达哥拉斯发现的 220 与 284 是最著名的一对，古人甚至相信刻有这对数的护身符能维系完美友情。此后两千多年未发现新对，直到费马、笛卡儿、欧拉等人陆续扩展列表；帕格尼尼发现了被忽略的 1184 与 1210。故事还介绍了“亲和链”以及尚未发现的 3 个数的亲和链——“众”。
中文正文
53° 亲和数
前面说过（故事 40），两个整数，如果每一个数都是另一个数的真因子之和，就是所谓的亲和数。例如，毕达哥拉斯发现的 220 和 284 就构成一对亲和数。这对特殊的数字还带着神秘的色彩。很多人相信，刻着这两个数字的护身符能让佩带它的人们永葆完美的友情。假如其中一个人受到了伤害，即使被针扎了一下，远在地球另一边的伙伴也能感觉。在魔法、巫术、占星和算命等活动中，这些数字扮演着重要的角色。
奇怪的是，以后再没发现新的亲和数。直到 1636 年，伟大的法国数论专家费马（Pierre Fermat）才宣布 17 296 与 18 416 结成另一对亲和数。两年后，法国数学家、哲学家笛卡儿（Descartes）发现了第三对。1747 年，瑞士数学家欧拉（Leonard Euler）系统研究了亲和数，推出了 30 对，然后又扩充到 60 多对。¹
寻找亲和数的历史上，还有一件奇怪的事情。1866 年，一个意大利少年帕格尼尼（Nicolo Paganini）发现了被忽略的相对较小的一对亲和数，1184 与 1210。今天，我们已经知道 900 多对亲和数。这些数对都有相同的奇偶性，就是说，同一对里的两个数要么同为奇数，要么同为偶数。所有奇数都是 3 的倍数，而所有偶数对的数字总和都是 9 的倍数。
三个或更多数的序列将形成一个“亲和链”，假如其相邻的两个数都构成亲和数对。我们只知道两个亲和链。1918 年，法国人普雷特（P. Proulet）发现了一条 5 个数的链：
12 496，14 288，15 472，14 536，14 264
还有 28 个数的链，起头一个数是 14 316。只有 3 个数的亲和链叫“众”，还没发现呢。
¹ 2005 年 3 月，人们发现了一对 24 073 位的亲和数。
英文正文
53° Amicable numbers. Two whole numbers, it will be recalled from Item 40°, are amicable if each is the sum of the proper divisors of the other. For example, 284 and 220, which constitute a pair that has been ascribed to Pythagoras, are amicable. This particular pair of numbers attained a mystical aura, and superstition maintained that two talismans bearing these numbers would seal perfect friendship between the wearers; should one of the wearers, though separated from the other by half the circumference of the earth, receive even so small an injury as a pin prick, the other would be aware of it. The numbers came to play an important role in magic, sorcery, astrology, and the casting of horoscopes.
Curiously enough, it seems that no new pair of amicable numbers was discovered until the great French number theorist Pierre de Fermat in 1636 announced 17,296 and 18,416 as another pair. Two years later the French mathematician and philosopher René Descartes gave a third pair. The Swiss mathematician Leonard Euler undertook a systematic search for amicable numbers and, in 1747, gave a list of thirty pairs, which he later extended to more than sixty.
A second curiosity in the history of these numbers was the late discovery, by the sixteen-year-old Italian boy Nicolo Paganini in 1866, of the overlooked and relatively small pair of amicable numbers, 1184 and 1210. Today more than nine hundred pairs of amicable numbers are known. These pairs are all of the same parity; that is, the two numbers of a pair are either both odd or both even. All the odd ones are multiples of 3; all the even pairs have the sum of their digits a multiple of 9.
A cyclic sequence of three or more numbers such that the sum of the proper divisors of each is equal to the next in the sequence is known as a sociable chain of numbers. Only two sociable chains are known. In 1918 the Frenchman P. Poulet found one of five "links":
12,496 14,288 15,472 14,536 14,264
There is a twenty-eight-link chain starting with 14,316. A sociable chain of exactly three links has been called a crowd; no crowds have yet been found.
英语学习拓展
常用词汇
amicable音标/ˈæmɪkəbl/
亲和的；友好的。
原文用法：Two whole numbers ... are amicable if each is the sum of the proper divisors of the other.
proper divisor音标/ˌprɒpə dɪˈvaɪzə(r)/
真因子（真约数）。
原文用法：同上。
ascribe音标/əˈskraɪb/
把……归于。
原文用法：a pair that has been ascribed to Pythagoras.
mystical音标/ˈmɪstɪkl/
神秘的。
原文用法：This particular pair of numbers attained a mystical aura...
aura音标/ˈɔːrə/
光环；氛围。
原文用法：同上。
superstition音标/ˌsuːpəˈstɪʃn/
迷信。
原文用法：superstition maintained that two talismans bearing these numbers...
talisman音标/ˈtælɪzmən/
护身符。
原文用法：同上。
seal音标/siːl/
封上；保证。
原文用法：would seal perfect friendship between the wearers.
wearer音标/ˈweərə(r)/
佩戴者。
原文用法：同上。
circumference音标/səˈkʌmfərəns/
圆周；周长。
原文用法：separated from the other by half the circumference of the earth.
pin prick音标/ˈpɪn prɪk/
针扎。
原文用法：receive even so small an injury as a pin prick.
aware音标/əˈweə(r)/
意识到的。
原文用法：the other would be aware of it.
sorcery音标/ˈsɔːsəri/
巫术。
原文用法：magic, sorcery, astrology, and the casting of horoscopes.
astrology音标/əˈstrɒlədʒi/
占星术。
原文用法：同上。
horoscope音标/ˈhɒrəskəʊp/
占星图；星座运势。
原文用法：the casting of horoscopes.
curiosity音标/ˌkjʊəriˈɒsəti/
好奇；奇事。
原文用法：A second curiosity in the history of these numbers...
announce音标/əˈnaʊns/
宣布。
原文用法：Pierre de Fermat in 1636 announced 17,296 and 18,416 as another pair.
undertook音标/ˌʌndəˈtʊk/
着手；承担（undertake 的过去式）。
原文用法：Leonard Euler undertook a systematic search...
systematic音标/ˌsɪstəˈmætɪk/
系统的。
原文用法：同上。
extend音标/ɪkˈstend/
扩展。
原文用法：which he later extended to more than sixty.
parity音标/ˈpærəti/
奇偶性。
原文用法：These pairs are all of the same parity.
multiple音标/ˈmʌltɪpl/
倍数。
原文用法：All the odd ones are multiples of 3.
sociable chain音标/ˈsəʊʃəbl tʃeɪn/
亲和链。
原文用法：a sociable chain of numbers.
crowd音标/kraʊd/
众（3 个数的亲和链）。
原文用法：A sociable chain of exactly three links has been called a crowd.
同义词与近义词
对应核心词汇：amicable
对应核心词汇：amicable。
辨析：amicable 指数学上的“亲和”，也可指友好关系；friendly 泛指友好。
对应核心词汇：proper divisor
对应核心词汇：proper divisor。
辨析：二者同义，proper divisor 更常见于数论文献。
对应核心词汇：ascribe
对应核心词汇：ascribe。
辨析：ascribe 常接 to，表示“把……归于”；attribute 更常用；credit 强调归功于。
对应核心词汇：talisman
对应核心词汇：talisman。
辨析：talisman 指护身符、辟邪物；charm 可指幸运符；amulet 也指护身符。
对应核心词汇：undertook
对应核心词汇：undertook。
辨析：undertake 强调“着手承担”；perform 侧重执行；conduct 侧重组织进行。
对应核心词汇：extend
对应核心词汇：extend。
辨析：extend 侧重延伸、扩展范围；expand 侧重扩大体积/规模；enlarge 侧重放大尺寸。
常用短语与固定搭配
- amicable numbers：亲和数。原文用法：Two whole numbers ... are amicable if ...
- proper divisors：真因子。原文用法：the sum of the proper divisors of the other.
- ascribed to：归于。原文用法：a pair that has been ascribed to Pythagoras.
- mystical aura：神秘光环。原文用法：attained a mystical aura.
- seal perfect friendship：维系完美友情。原文用法：would seal perfect friendship between the wearers.
- separated by half the circumference of the earth：相隔半个地球。原文用法：though separated from the other by half the circumference of the earth.
- pin prick：针扎。原文用法：receive even so small an injury as a pin prick.
- play an important role in：在……中扮演重要角色。原文用法：The numbers came to play an important role in magic, sorcery, astrology...
- casting of horoscopes：占星算命。原文用法：and the casting of horoscopes.
- undertook a systematic search：进行系统搜索。原文用法：undertook a systematic search for amicable numbers.
- extend to：扩展到。原文用法：extended to more than sixty.
- of the same parity：奇偶性相同。原文用法：all of the same parity.
- multiples of 9：9 的倍数。原文用法：the sum of their digits a multiple of 9.
- sociable chain of numbers：亲和链。原文用法：known as a sociable chain of numbers.
- no crowds have yet been found：尚未发现“众”。原文用法：no crowds have yet been found.
常用句式
- Pattern: A and B are X if each is the ... of the other.：如果 A 和 B 各为对方的……，则它们是 X。原文例句：Two whole numbers, it will be recalled from Item 40°, are amicable if each is the sum of the proper divisors of the other.
- Pattern: A is known as B.：A 被称为 B。原文例句：A cyclic sequence of three or more numbers ... is known as a sociable chain of numbers.
- Pattern: No ... have yet been found.：尚未发现……。原文例句：no crowds have yet been found.
- Pattern: should one of ..., ... would ...：虚拟条件句，表示“即使……也……”。原文例句：should one of the wearers, though separated from the other by half the circumference of the earth, receive even so small an injury as a pin prick, the other would be aware of it.
长难句解析
This particular pair of numbers attained a mystical aura, and superstition maintained that two talismans bearing these numbers would seal perfect friendship between the wearers; should one of the wearers, though separated from the other by half the circumference of the earth, receive even so small an injury as a pin prick, the other would be aware of it.
- 主干：This pair attained a mystical aura, and superstition maintained that two talismans would seal perfect friendship.
- 修饰与从句：
- that two talismans ... would seal perfect friendship 为宾语从句； - bearing these numbers 为现在分词短语，修饰 talismans； - should one of the wearers ... receive ... injury 为虚拟条件倒装句（= if one of the wearers should receive ...）； - though separated from the other by half the circumference of the earth 为让步状语； - as a pin prick 为比较结构，说明伤害之小。
- 逻辑关系：前半句说明 220 与 284 的神秘色彩，后半句用虚拟条件夸张地说明佩戴这对数护身符的两人心灵相通。
语言使用特点
- 定义句清晰严谨，使用 if each is the sum of ... 的数学表达；
- 历史叙述按时间顺序展开，使用过去时；
- 用 should ... would ... 虚拟倒装表达夸张假设；
- 数字表达规范，注意英文千位逗号与中文空格的差异。
迁移练习
- 通过列出真因子验证 220 与 284 是一对亲和数。
- 用英语解释什么是“亲和链”（sociable chain）。
- 翻译：今天，我们已经知道 900 多对亲和数。
点击展开参考答案
- （示例）The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, and their sum is 284. The proper divisors of 284 are 1, 2, 4, 71, 142, and their sum is 220. Therefore 220 and 284 are amicable.
- （示例）A sociable chain is a cyclic sequence of three or more numbers in which the sum of the proper divisors of each number equals the next number in the sequence.
- Today more than nine hundred pairs of amicable numbers are known.