# Adrien-Marie Legendre

> French mathematician (1752–1833)

**Wikidata**: [Q191021](https://www.wikidata.org/wiki/Q191021)  
**Wikipedia**: [English](https://en.wikipedia.org/wiki/Adrien-Marie_Legendre)  
**Source**: https://4ort.xyz/entity/adrien-marie-legendre

## Summary
Adrien-Marie Legendre was a French mathematician (1752–1833) known for his foundational contributions to number theory, including the development of Legendre polynomials, the Legendre symbol, and the Legendre transformation. He was a member of the French Academy of Sciences and made significant advancements in pure mathematics, particularly in the study of integers and algebraic equations.

## Biography
- **Born**: September 18, 1752, in Paris, France
- **Nationality**: French
- **Education**: Studied at the Collège Mazarin and later at the École Normale Supérieure
- **Known for**: Contributions to number theory, including Legendre polynomials, the Legendre symbol, and the Legendre transformation
- **Employer(s)**: French Academy of Sciences, Bureau des Longitudes
- **Field(s)**: Number theory, pure mathematics

## Contributions
Adrien-Marie Legendre made several key contributions to mathematics, particularly in number theory:
- **Legendre Polynomials**: Developed solutions to Legendre's differential equation, which are fundamental in mathematical analysis and physics.
- **Legendre Symbol**: Introduced a multiplicative function with values 1, −1, or 0, crucial for understanding quadratic residues in number theory.
- **Legendre Transformation**: Formulated an involutive transformation on real-valued convex functions, impacting thermodynamics and optimization.
- **Legendre's Conjecture**: Proposed that there is always a prime number between any two square numbers, though it remains unproven.
- **Legendre's Theorem**: Established conditions for the existence of solutions to the Diophantine equation \( ax^2 + by^2 + cz^2 = 0 \).
- **Legendre's Formula**: Contributed to number theory with an expression named after him.
- **Legendre's Theorem on Spherical Triangles**: Provided a geometric theorem in spherical geometry.

## FAQs
**What are Adrien-Marie Legendre's most significant mathematical contributions?**
Legendre is best known for his work on Legendre polynomials, the Legendre symbol, and the Legendre transformation, which have profound applications in mathematics, physics, and engineering.

**Where did Adrien-Marie Legendre work and study?**
He studied at the Collège Mazarin and later at the École Normale Supérieure. He was affiliated with the French Academy of Sciences and the Bureau des Longitudes.

**What is the Legendre symbol, and why is it important?**
The Legendre symbol is a multiplicative function that determines whether a given integer is a quadratic residue modulo a prime number. It is essential in number theory and cryptography.

**What is the Legendre transformation, and how is it used?**
The Legendre transformation is an involutive transformation on real-valued convex functions, widely used in thermodynamics, optimization, and control theory.

**What is Legendre's conjecture, and is it proven?**
Legendre's conjecture states that there is always a prime number between any two square numbers. While widely believed, it remains unproven as of now.

## Why They Matter
Adrien-Marie Legendre's work in number theory and mathematical analysis laid the groundwork for modern algebraic structures and differential equations. His contributions to Legendre polynomials and the Legendre symbol are foundational in fields such as physics, engineering, and cryptography. The Legendre transformation remains a cornerstone of thermodynamics and optimization, demonstrating the enduring impact of his mathematical innovations. Legendre's legacy continues to influence contemporary research in pure and applied mathematics.

## Notable For
- **Foundational Contributions to Number Theory**: Developed Legendre polynomials, the Legendre symbol, and the Legendre transformation.
- **Legendre's Conjecture**: Proposed a significant unproven conjecture in number theory.
- **Legendre's Theorem**: Established conditions for solutions to the Diophantine equation \( ax^2 + by^2 + cz^2 = 0 \).
- **Member of the French Academy of Sciences**: Affiliated with one of France's most prestigious scientific institutions.
- **Legendre's Formula**: Contributed to number theory with a named expression.
- **Legendre's Theorem on Spherical Triangles**: Provided a key result in spherical geometry.

## Body

### Early Life and Education
Adrien-Marie Legendre was born on September 18, 1752, in Paris, France. He attended the Collège Mazarin before pursuing higher education at the École Normale Supérieure. His early mathematical aptitude was evident, and he began contributing to the field of number theory at a young age.

### Academic Career and Affiliations
Legendre's academic career was marked by his affiliation with the French Academy of Sciences and the Bureau des Longitudes. He held positions at these institutions, where he conducted research and published his findings. His work was recognized for its rigor and originality, earning him respect within the mathematical community.

### Mathematical Contributions
Legendre's mathematical contributions are extensive and influential:
- **Legendre Polynomials**: He developed solutions to Legendre's differential equation, which are now fundamental in mathematical analysis and physics. These polynomials are named after him and are widely used in solving partial differential equations.
- **Legendre Symbol**: The Legendre symbol is a multiplicative function that determines whether an integer is a quadratic residue modulo a prime number. It is crucial in number theory and has applications in cryptography.
- **Legendre Transformation**: This transformation is used in thermodynamics and optimization, allowing for the conversion between different forms of energy and variables. It is named after Legendre and is essential in various scientific disciplines.
- **Legendre's Conjecture**: He proposed that there is always a prime number between any two square numbers, though this conjecture remains unproven. It has significant implications for the distribution of prime numbers.
- **Legendre's Theorem**: He established conditions for the existence of solutions to the Diophantine equation \( ax^2 + by^2 + cz^2 = 0 \), contributing to the study of integer solutions to quadratic forms.
- **Legendre's Formula**: He contributed to number theory with an expression named after him, which has applications in various mathematical contexts.
- **Legendre's Theorem on Spherical Triangles**: This theorem provides a key result in spherical geometry, which is used in navigation and astronomy.

### Legacy and Influence
Adrien-Marie Legendre's legacy extends beyond his lifetime, with his mathematical contributions still being studied and applied today. His work on Legendre polynomials and the Legendre symbol has found applications in physics, engineering, and cryptography. The Legendre transformation remains a fundamental concept in thermodynamics and optimization. His conjecture on prime numbers continues to inspire research in number theory. Legendre's influence is evident in the ongoing development of mathematical theories and their practical applications.

## References

1. Great Soviet Encyclopedia (1969–1978)
2. MacTutor History of Mathematics archive
3. Bureau des Longitudes
4. [Auteuil Cemetery](https://commons.wikimedia.org/wiki/File:Tombe_Adrien-Marie_Legendre,_Cimetière_d%27Auteuil,_Paris.jpg)
5. BnF authorities
6. CERL Thesaurus
7. Encyclopædia Britannica
8. Integrated Authority File
9. Library of the World's Best Literature
10. Find a Grave
11. [Source](https://cdn.paris.fr/paris/2022/06/13/21cfa8169a72caecf81fb3b3ad71c227.pdf)
12. [Source](https://www.toureiffel.paris/fr/le-monument/tour-eiffel-et-sciences)
13. International Standard Name Identifier
14. CiNii Research
15. [Book of Members, 1780–2010: Chapter L (PDF). American Academy of Arts and Sciences](http://www.amacad.org/publications/BookofMembers/ChapterL.pdf)
16. SNAC
17. Gran Enciclopèdia Catalana
18. Croatian Encyclopedia
19. La France savante
20. Freebase Data Dumps. 2013
21. Virtual International Authority File
22. [Source](http://digitale.beic.it/primo_library/libweb/action/search.do?fn=search&vid=BEIC&vl%283134987UI0%29=creator&vl%28freeText0%29=Legendre%20Adrien-Marie)
23. [Source](https://www.bartleby.com/library/bios/index10.html)
24. [BnF authorities](http://data.bnf.fr/ark:/12148/cb12443375d)
25. Autoritats UB
26. Quora
27. Catalogo of the National Library of India