Lagrange's four-square theorem
Teorema de los cuatro cuadrados de Lagrange
- Date
- 1770 · confidence high
- Place
- Prussian Academy of Sciences, Berlin, Germany
- Credited to
- Joseph-Louis Lagrange
- Field
- Knowledge & symbols
What was discovered
Lagrange proved that every nonnegative integer can be written as the sum of at most four perfect squares, settling a conjecture that Bachet had noted in 1621 when translating Diophantus's 'Arithmetica', already suspected by Fermat and Euler.
Diophantus already knew cases of the result in antiquity, and Bachet conjectured it in 1621; Lagrange was the one who proved it in 1770, the same year Waring posed his general problem.
What it led to
- Legendre's three-square theorem (1797-1798)
- Jacobi's formula for the number of representations as a sum of four squares (1834)
- A boost to Waring's problem on sums of k-th powers
Built on
- Bachet's conjecture on the sum of four squares (1621)