amicable number: meaning, definition, pronunciation and examples
Very lowTechnical (mathematical)
Quick answer
What does “amicable number” mean?
A pair of numbers where each is the sum of the proper divisors of the other.
Audio
Pronunciation
Definition
Meaning and Definition
A pair of numbers where each is the sum of the proper divisors of the other.
In number theory, a pair of distinct positive integers (m, n) such that the sum of the proper divisors of m equals n, and the sum of the proper divisors of n equals m. The smallest amicable pair is (220, 284).
Dialectal Variation
British vs American Usage
Differences
No significant linguistic differences. The term is standardised in global mathematical discourse.
Connotations
Identical connotations: purely technical, without emotional or figurative nuance.
Frequency
Exceedingly rare outside mathematics textbooks or specialised discussions. No frequency difference between varieties.
Grammar
How to Use “amicable number” in a Sentence
The numbers X and Y are amicable.Z is one half of an amicable pair.Vocabulary
Collocations
Examples
Examples of “amicable number” in a Sentence
adjective
British English
- The numbers 1184 and 1210 form an amicable pair.
- Fermat discovered an amicable set.
American English
- 1184 and 1210 are an amicable pair.
- He researched amicable number sequences.
Usage
Meaning in Context
Business
Never used.
Academic
Used exclusively in pure mathematics, number theory, or the history of mathematics.
Everyday
Virtually never used.
Technical
Core term in recreational and theoretical number theory.
Vocabulary
Synonyms of “amicable number”
Neutral
Weak
Vocabulary
Antonyms of “amicable number”
Watch out
Common Mistakes When Using “amicable number”
- Using 'amicable' as a general adjective to describe numbers (e.g., 'The figures were amicable').
- Confusing amicable numbers with perfect numbers (where the sum of divisors equals the number itself, e.g., 6).
- Assuming the term has any relation to social or diplomatic contexts.
FAQ
Frequently Asked Questions
The smallest and most famous pair is (220, 284).
They were known to the Pythagoreans, but the formula for generating some pairs is attributed to Thābit ibn Qurra (9th century).
No, they are quite rare. Over 12 million pairs are known, but this is a tiny fraction of all integers.
A perfect number equals the sum of its own proper divisors (e.g., 6). Amicable numbers are a pair of different numbers where each is the sum of the other's proper divisors.
A pair of numbers where each is the sum of the proper divisors of the other.
Amicable number is usually technical (mathematical) in register.
Amicable number: in British English it is pronounced /ˌæm.ɪ.kə.bəl ˈnʌm.bər/, and in American English it is pronounced /ˈæm.ɪ.kə.bəl ˈnʌm.bɚ/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think: Amicable numbers are like perfect mathematical friends. 220 gives its divisors (1+2+4+5+10+11+20+22+44+55+110) to 284, and 284 returns the favour (1+2+4+71+142 to 220). They complete each other.
Conceptual Metaphor
MATHEMATICAL RELATIONSHIP IS FRIENDSHIP. A specialised metaphor from the etymology of 'amicable' (friendly), frozen in this technical term.
Practice
Quiz
What defines a pair of amicable numbers?