= Lagrange's four-square theorem
{c}
{title2=$n=a^2+b^2+c^2+d^2$}
{wiki}
= Lagrange four-square theorem
{c}
{synonym}
\b[Every <nonnegative integer> is a <sum of squares> with four terms, with zero terms allowed.] Precisely, for every <integer> $n\geq0$, there exist <nonnegative integers> $a,b,c,d$ such that
$$
n=a^2+b^2+c^2+d^2.
$$
Allowing arbitrary <integer> roots gives the same statement: changing a root to its <absolute value> leaves the resulting <square number> unchanged. Four is a uniform upper bound on the number of nonzero terms, rather than a demand for four positive terms. Repeated <square numbers> are allowed. Conrad states the positive case and explicitly explains the role of zero terms; $n=0$ follows by taking all four roots zero. https://kconrad.math.uconn.edu/blurbs/ugradnumthy/descent.pdf[Keith Conrad, Proofs by Descent, Section 6, opening paragraph and Theorem 6.6]{ref}
The small cases make the convention visible:
$$
\begin{aligned}
0&=0^2+0^2+0^2+0^2,\\
1&=1^2+0^2+0^2+0^2,\\
2&=1^2+1^2+0^2+0^2,\\
3&=1^2+1^2+1^2+0^2,\\
7&=2^2+1^2+1^2+1^2.
\end{aligned}
$$
In particular, $1$ could not be a <sum of squares> with four strictly positive roots, since each term would be at least one. Geometrically, a <square> of <integer> side length $a$ has <area> $a^2$: the theorem says that any nonnegative whole-number total area can be split into at most four such square areas. It does not assert that those <squares> tile one larger <square> or a prescribed <rectangle>.
\Image[/four-square-areas.png]
{title=Four-square representations as areas counted in unit cells}
{description=The roots for $7,23,28,31$ were found by enumerating nonnegative four-tuples. Each panel has the same unit length, and each little cell has <area> one. The four separate <squares> illustrate addition of areas, with no claim about tiling a prescribed shape. These are exact arithmetic examples, not measured data.}
Why does the fourth term matter? Two <square numbers> do not suffice for $3$: in <modular arithmetic>, a <square number> has <remainder> zero or one modulo four, so their sum cannot be three modulo four. The <sum of two squares theorem> gives a complete criterion for the positive <integers> representable with two terms. Even three terms leave gaps. A <square number> has <remainder> $0,1,$ or $4$ modulo eight; adding three of these, with repetition allowed, never gives seven modulo eight. Thus $7,15,23,31,\ldots$ cannot be sums of three <square numbers>, although <Lagrange's four-square theorem> represents all of them with four. This also proves that the bound four cannot be reduced to three.
The modulo-eight obstruction is not the only way that three terms can fail. For example, $28$ needs four: if three <square numbers> added to $28$, their <remainders> modulo four would force all three roots even. Dividing by four would then express $7$ as a sum of three <square numbers>, contradicting the preceding calculation. These short checks explain the examples without requiring a full classification of sums of three <square numbers>.
\Image[/four-square-minima.png]
{title=Minimum number of nonzero square terms for integers from one to one hundred}
{description=Each cell is labelled by the <positive integer> being represented; its colour records the least number of nonzero <square numbers> whose sum equals it. Direct enumeration gives $10$ requiring one term, $33$ two, $42$ three, and $15$ four. Zero is omitted from the grid and requires no nonzero terms. Zero padding converts every representation into four slots. This finite computation illustrates <Lagrange's four-square theorem>; it does not prove its assertion for all <nonnegative integers>.}
The practical benefit of <Lagrange's four-square theorem> is an existence guarantee for a family of <Diophantine equations>: for every <nonnegative integer> $n$, the displayed <equation> has an <integer> solution. It does not specify a unique solution. For instance, $65=8^2+1^2+0^2+0^2=7^2+4^2+0^2+0^2=6^2+4^2+3^2+2^2$. To find a representation by a finite search, enumerate $0\leq d\leq c\leq b\leq a\leq\lfloor\sqrt n\rfloor$ and test the sum. The <floor function> supplies a finite bound because no single term can exceed $n$, and sorting roots removes duplicate orderings without discarding any possible sum. The theorem guarantees that the search succeeds.
A <greedy algorithm> that repeatedly subtracts the largest available <square number> need not find four terms: for $23$ it chooses $16+4+1+1+1$, while $23=9+9+4+1$ uses four. The appeal of the theorem is this gap between local choices and a global promise: individual <square numbers> become farther apart, and two or three terms leave arithmetic obstructions, yet four terms cover every <nonnegative integer>.
For historical orientation, Conrad dates <Lagrange's four-square theorem> to $1770$ and identifies the memoir as \i[Démonstration d'un théorème d'arithmétique]: it appeared in the Berlin Academy's volume for the year $1770$, published in $1772$, pages $123$–$133$. Keeping the volume year and publication year distinct avoids an apparent conflict between dates. https://kconrad.math.uconn.edu/blurbs/ugradnumthy/descent.pdf[Keith Conrad, Proofs by Descent, Theorem 6.6 and reference 11]{ref}
Back to article page
