diophantine equation: meaning, definition, pronunciation and examples
C2Formal, Academic, Technical
Quick answer
What does “diophantine equation” mean?
A polynomial equation with integer coefficients that is to be solved for integer values.
Audio
Pronunciation
Definition
Meaning and Definition
A polynomial equation with integer coefficients that is to be solved for integer values.
In mathematics, a Diophantine equation is any algebraic equation in one or more unknowns whose solutions are sought in integers or, more broadly, rational numbers. They are named after the 3rd-century Alexandrian mathematician Diophantus. Solving them often involves number theory techniques.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or usage.
Connotations
Identical academic connotations.
Frequency
Equally rare and specialised in both varieties.
Grammar
How to Use “diophantine equation” in a Sentence
[Subject: mathematician/algorithm] + solve(s) + [Object: diophantine equation][Subject: problem] + is/becomes + [Complement: a diophantine equation][Preposition: of] + diophantine equations + [Verb: constitute/form]Vocabulary
Collocations
Examples
Examples of “diophantine equation” in a Sentence
adjective
British English
- The Diophantine problem remained unsolved for centuries.
- He specialised in Diophantine analysis.
American English
- The Diophantine problem remained unsolved for centuries.
- She specialized in Diophantine analysis.
Usage
Meaning in Context
Business
Virtually never used.
Academic
Exclusively used in advanced mathematics, particularly in number theory, algebraic geometry, and cryptography papers.
Everyday
Never used in everyday conversation.
Technical
Used precisely in mathematical research, computer algebra systems, and advanced algorithm design (e.g., for cryptographic protocols).
Vocabulary
Synonyms of “diophantine equation”
Neutral
Weak
Vocabulary
Antonyms of “diophantine equation”
Watch out
Common Mistakes When Using “diophantine equation”
- Mispronouncing as 'dee-oh-fan-teen' or 'dye-oh-fan-tin'.
- Using it to refer to any difficult equation.
- Misspelling as 'Diophantine' without the 'i' after 'Diophant'.
FAQ
Frequently Asked Questions
No, it is a highly specialised term used almost exclusively in advanced mathematics and theoretical computer science.
Yes, many Diophantine equations have no integer solutions (e.g., x² + y² = 3). Proving the absence of solutions is a major part of the study.
A linear example is 2x + 3y = 1. A famous nonlinear example is xⁿ + yⁿ = zⁿ (related to Fermat's Last Theorem).
Yes, because it is derived from the proper name Diophantus. It is always capitalized.
Diophantine equation is usually formal, academic, technical in register.
Diophantine equation: in British English it is pronounced /ˌdʌɪə(ʊ)ˈfantɪn ɪˈkweɪʒ(ə)n/, and in American English it is pronounced /ˌdaɪəˈfæntin ɪˈkweɪʒən/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think: 'Die-o-FAN-tine' – Imagine the ancient mathematician Diophantus 'fainting' because finding INTEGER solutions to his equations was so hard.
Conceptual Metaphor
A puzzle box with only whole-number locks. (The solutions must be integers, not fractions or decimals.)
Practice
Quiz
What is the defining feature of a Diophantine equation?