Editing Lagrange's four-square theorem

{{short description|Every natural number can be represented as the sum of four integer squares}}
{{for multi|Lagrange's identity|Lagrange's identity (disambiguation)|Lagrange's theorem|Lagrange's theorem (disambiguation)}}
{{Redirect2|four-square theorem|four square theorem|other uses|four square (disambiguation)}}
[[File:distances_between_double_cube_corners.svg|thumb|Unlike in three dimensions in which distances between [[Vertex (geometry)|vertices]] of a [[polycube]] with unit edges excludes √7 due to [[Legendre's three-square theorem]], Lagrange's four-square theorem states that the analogue in four dimensions yields [[square root]]s of every [[natural number]] ]]

'''Lagrange's four-square theorem''', also known as '''Bachet's conjecture''', states that every [[natural number|nonnegative integer]] can be represented as a sum of four non-negative integer [[square number|squares]].{{r|andrews}} That is, the squares form an [[additive basis]] of order four:
<math display="block">p = a^2 + b^2 + c^2 + d^2,</math> 
where the four numbers <math>a, b, c, d</math> are integers. For illustration, 3, 31, and 310 can be represented as the sum of four squares as follows:
<math display="block">\begin{align}
3 & = 1^2+1^2+1^2+0^2 \\[3pt]
31 & = 5^2+2^2+1^2+1^2 \\[3pt]
310 & = 17^2+4^2+2^2+1^2 \\[3pt]
& = 16^2 + 7^2 + 2^2 +1^2 \\[3pt]
& = 15^2 + 9^2 + 2^2 +0^2 \\[3pt]
& = 12^2 + 11^2 + 6^2 + 3^2.
\end{align}</math>

This theorem was proven by [[Joseph Louis Lagrange]] in 1770. It is a special case of the [[Fermat polygonal number theorem]].

==Historical development==
From examples given in the ''[[Arithmetica]],'' it is clear that [[Diophantus]] was aware of the theorem. This book was translated in 1621 into Latin by [[Claude Gaspard Bachet de Méziriac|Bachet (Claude Gaspard Bachet de Méziriac)]], who stated the theorem in the notes of his translation. But the theorem was not proved until 1770 by Lagrange.<ref>{{harvnb|Ireland|Rosen|1990}}.</ref>

[[Adrien-Marie Legendre]] extended the theorem in 1797–8 with his [[Legendre's three-square theorem|three-square theorem]], by proving that a positive integer can be expressed as the sum of three squares if and only if it is not of the form <math>4^k(8m+7)</math> for integers {{mvar|k}} and {{mvar|m}}. Later, in 1834, [[Carl Gustav Jakob Jacobi]] discovered a simple formula for the number of representations of an integer as the sum of four squares with his own [[Jacobi's four-square theorem|four-square theorem]].

The formula is also linked to [[Descartes' theorem]] of four "kissing circles", which involves the sum of the squares of the curvatures of four circles. This is also linked to [[Apollonian gasket]]s, which were more recently related to the [[Ramanujan–Petersson conjecture]].<ref>{{harvnb|Sarnak|2013}}.</ref>

== Proofs ==

=== The classical proof ===
Several very similar modern versions<ref>{{harvnb|Landau|1958|loc=Theorems 166 to 169}}.</ref><ref>{{harvnb|Hardy|Wright|2008|loc=Theorem 369}}.</ref><ref>{{harvnb|Niven|Zuckerman|1960|loc=paragraph 5.7}}.</ref> of Lagrange's proof exist. The proof below is a slightly simplified version, in which the cases for which ''m'' is even or odd do not require separate arguments.

{{math proof|title=The classical proof|proof=
It is sufficient to prove the theorem for every odd prime number ''p''. This immediately follows from [[Euler's four-square identity]] (and from the fact that the theorem is true for the numbers 1 and 2).

The residues of ''a''<sup>2</sup> modulo ''p'' are distinct for every ''a'' between 0 and {{math|(''p'' − 1)/2}} (inclusive).
To see this, take some ''a'' and define ''c'' as ''a''<sup>2</sup> mod ''p''.
''a'' is a root of the polynomial {{math|''x''<sup>2</sup> − ''c''}} over the field {{math|[[Finite field|Z/''p''Z]]}}.
So is {{math|''p'' − ''a''}} (which is different from ''a'').
In a field ''K'', any polynomial of degree ''n'' has at most ''n'' distinct roots ([[Lagrange's theorem (number theory)]]),
so there are no other ''a'' with this property, in particular not among 0 to {{math|(''p'' − 1)/2}}.

Similarly, for ''b'' taking integral values between 0 and {{math|(''p'' − 1)/2}} (inclusive), the {{math|−''b''<sup>2</sup> − 1}} are distinct.
By the [[pigeonhole principle]], there are ''a'' and ''b'' in this range, for which ''a''<sup>2</sup> and {{math|−''b''<sup>2</sup> − 1}} are congruent modulo ''p'', that is for which
<math display="block">a^2 + b^2 + 1^2 + 0^2 = np.</math>

Now let ''m'' be the smallest positive integer such that ''mp'' is the sum of four squares, {{math|''x''<sub>1</sub><sup>2</sup> + ''x''<sub>2</sub><sup>2</sup> + ''x''<sub>3</sub><sup>2</sup> + ''x''<sub>4</sub><sup>2</sup>}} (we have just shown that there is some ''m'' (namely ''n'') with this property, so there is a least one ''m'', and it is smaller than ''p''). We show by contradiction that ''m'' equals 1: supposing it is not the case, we prove the existence of a positive integer ''r'' less than ''m'', for which ''rp'' is also the sum of four squares (this is in the spirit of the [[proof by infinite descent|infinite descent]]<ref>Here the argument is a direct [[proof by contradiction]]. With the initial assumption that ''m'' > 2, ''m'' < ''p'', is ''some'' integer such that ''mp'' is the sum of four squares (not necessarily the smallest), the argument could be modified to become an infinite descent argument in the spirit of Fermat.</ref> method of Fermat).

For this purpose, we consider for each ''x''<sub>''i''</sub> the ''y''<sub>''i''</sub> which is in the same residue class modulo ''m'' and between {{math|(–''m'' + 1)/2}} and ''m''/2 (possibly included). It follows that {{math|1=''y''<sub>1</sub><sup>2</sup> + ''y''<sub>2</sub><sup>2</sup> + ''y''<sub>3</sub><sup>2</sup> + ''y''<sub>4</sub><sup>2</sup> = ''mr''}}, for some strictly positive integer ''r'' less than&nbsp;''m''.

Finally, another appeal to Euler's four-square identity shows that {{math|1=''mpmr'' = ''z''<sub>1</sub><sup>2</sup> + ''z''<sub>2</sub><sup>2</sup> + ''z''<sub>3</sub><sup>2</sup> + ''z''<sub>4</sub><sup>2</sup>}}. But the fact that each ''x''<sub>''i''</sub> is congruent to its corresponding ''y''<sub>''i''</sub> implies that all of the ''z''<sub>''i''</sub> are divisible by ''m''. Indeed,
<math display="block">\begin{cases}
   z_1 &= x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4 &\equiv x_1^2 + x_2^2 + x_3^2 + x_4^2 &= mp \equiv 0 &\pmod{m}, \\
   z_2 &= x_1 y_2 - x_2 y_1 + x_3 y_4 - x_4 y_3 &\equiv x_1 x_2 - x_2 x_1 + x_3 x_4 - x_4 x_3 &= 0 &\pmod{m}, \\
   z_3 &= x_1 y_3 - x_2 y_4 - x_3 y_1 + x_4 y_2 &\equiv x_1 x_3 - x_2 x_4 - x_3 x_1 + x_4 x_2 &= 0 &\pmod{m}, \\
   z_4 &= x_1 y_4 + x_2 y_3 - x_3 y_2 - x_4 y_1 &\equiv x_1 x_4 + x_2 x_3 - x_3 x_2 - x_4 x_1 &= 0 &\pmod{m}.
 \end{cases}</math>

It follows that, for {{math|1=''w''<sub>''i''</sub> = ''z''<sub>''i''</sub>/''m''}}, {{math|1=''w''<sub>1</sub><sup>2</sup> + ''w''<sub>2</sub><sup>2</sup> + ''w''<sub>3</sub><sup>2</sup> + ''w''<sub>4</sub><sup>2</sup> = ''rp''}}, and this is in contradiction with the minimality of&nbsp;''m''.

In the descent above, we must rule out both the case {{math|1=''y''<sub>1</sub> = ''y''<sub>2</sub> = ''y''<sub>3</sub> = ''y''<sub>4</sub> = ''m''/2}} (which would give {{math|1=''r'' = ''m''}} and no descent), and also the case {{math|1=''y''<sub>1</sub> = ''y''<sub>2</sub> = ''y''<sub>3</sub> = ''y''<sub>4</sub> = 0}} (which would give {{math|1=''r'' = 0}} rather than strictly positive).  For both of those cases, one can check that {{math|1=''mp'' = ''x''<sub>1</sub><sup>2</sup> + ''x''<sub>2</sub><sup>2</sup> + ''x''<sub>3</sub><sup>2</sup> + ''x''<sub>4</sub><sup>2</sup>}} would be a multiple of ''m''<sup>2</sup>, contradicting the fact that ''p'' is a prime greater than ''m''.
}}

=== Proof using the Hurwitz integers ===
Another way to prove the theorem relies on [[Hurwitz quaternion]]s, which are the analog of [[integer]]s for [[quaternion]]s.<ref name="Stillwell_2003">{{harvnb|Stillwell|2003|pages=138–157}}.</ref> 

{{math proof|title=Proof using the Hurwitz integers|proof=
The Hurwitz quaternions consist of all quaternions with integer components and all quaternions with [[half-integer]] components. These two sets can be combined into a single formula
<math display="block">\alpha = \frac{1}{2} E_0 (1 + \mathbf{i} + \mathbf{j} + \mathbf{k}) +E_1\mathbf{i} +E_2\mathbf{j} + E_3\mathbf{k} = a_0 +a_1\mathbf{i} +a_2\mathbf{j} +a_3\mathbf{k}</math>
where <math>E_0, E_1, E_2, E_3</math> are integers. Thus, the quaternion components <math>a_0, a_1, a_2, a_3</math> are either all integers or all half-integers, depending on whether <math>E_0</math> is even or odd, respectively. The set of Hurwitz quaternions forms a [[ring (mathematics)|ring]]; that is to say, the sum or product of any two Hurwitz quaternions is likewise a Hurwitz quaternion.

The [[field norm|(arithmetic, or field) norm]] <math>\mathrm N(\alpha)</math> of a rational quaternion <math>\alpha</math> is the nonnegative [[rational number]]
<math display="block">\mathrm{N}(\alpha) = \alpha\bar\alpha = a_0^2 + a_1^2 + a_2^2 + a_3^2</math>
where <math>\bar\alpha=a_0 -a_1\mathbf{i} -a_2\mathbf{j} -a_3\mathbf{k}</math> is the [[quaternion#Conjugation, the norm, and reciprocal|conjugate]] of <math>\alpha</math>. Note that the norm of a Hurwitz quaternion is always an integer. (If the coefficients are half-integers, then their squares are of the form <math>\tfrac{1}{4} + n : n \in \mathbb{Z}</math>, and the sum of four such numbers is an integer.)

Since quaternion multiplication is associative, and real numbers commute with other quaternions, the norm of a product of quaternions equals the product of the norms:
<math display="block"> \mathrm{N}(\alpha\beta)=\alpha\beta(\overline{\alpha\beta})=\alpha\beta\bar{\beta}\bar{\alpha}=\alpha \mathrm{N}(\beta)\bar\alpha=\alpha\bar\alpha \mathrm{N}(\beta)= \mathrm{N}(\alpha) \mathrm{N}(\beta).</math>

For any <math>\alpha\ne0</math>, <math>\alpha^{-1}=\bar\alpha\mathrm N(\alpha)^{-1}</math>. It follows easily that <math>\alpha</math> is a unit in the ring of Hurwitz quaternions if and only if <math>\mathrm N(\alpha)=1</math>.

The proof of the main theorem begins by reduction to the case of prime numbers. [[Euler's four-square identity]] implies that if Lagrange's four-square theorem holds for two numbers, it holds for the product of the two numbers. Since any natural number can be factored into powers of primes, it suffices to prove the theorem for prime numbers. It is true for <math>2 = 1^2 + 1^2 + 0^2 + 0^2</math>. To show this for an odd prime integer {{mvar|p}}, represent it as a quaternion <math>(p,0,0,0)</math> and assume for now (as we shall show later) that it is not a Hurwitz [[irreducible element|irreducible]]; that is, it can be factored into two non-unit Hurwitz quaternions
<math display="block">p = \alpha\beta.</math>

The norms of <math>p,\alpha,\beta</math> are integers such that
<math display="block">\mathrm N(p)=p^2=\mathrm N(\alpha\beta)=\mathrm N(\alpha)\mathrm N(\beta)</math>
and <math>\mathrm N(\alpha),\mathrm N(\beta) > 1</math>. This shows that both <math>\mathrm N(\alpha)</math> and <math>\mathrm N(\beta)</math> are equal to {{mvar|p}} (since they are integers), and {{mvar|p}} is the sum of four squares
<math display="block">p=\mathrm N(\alpha)=a_0^2+a_1^2+a_2^2+a_3^2.</math>

If it happens that the <math>\alpha</math> chosen has half-integer coefficients, it can be replaced by another Hurwitz quaternion. Choose <math>\omega = (\pm 1\pm\mathbf{i}\pm\mathbf{j} \pm\mathbf{k})/2</math> in such a way that <math>\gamma \equiv \omega + \alpha</math> has even integer coefficients. Then
<math display="block">p=(\bar\gamma-\bar\omega)\omega\bar\omega(\gamma-\omega)=(\bar\gamma\omega-1)(\bar\omega\gamma-1).</math>

Since <math>\gamma</math> has even integer coefficients, <math>(\bar\omega\gamma-1)</math> will have integer coefficients and can be used instead of the original <math>\alpha</math> to give a representation of {{mvar|p}} as the sum of four squares.

As for showing that {{mvar|p}} is not a Hurwitz irreducible, [[Joseph Louis Lagrange|Lagrange]] proved that any odd prime {{mvar|p}} divides at least one number of the form <math>u=1+l^2+m^2</math>, where {{mvar|l}} and {{mvar|m}} are integers.<ref name="Stillwell_2003" /> This can be seen as follows: since {{mvar|p}} is prime, <math>a^2\equiv b^2\pmod p</math> can hold for integers <math>a,b</math>, only when <math>a\equiv\pm b\pmod p</math>. Thus, the set <math>X=\{0^2,1^2,\dots,((p-1)/2)^2\}</math> of squares contains <math>(p+1)/2</math> distinct [[modular arithmetic#Congruence classes|residues]] modulo {{mvar|p}}. Likewise, <math>Y=\{-(1+x):x\in X\}</math> contains <math>(p+1)/2</math> residues. Since there are only {{mvar|p}} residues in total, and <math>|X|+|Y| = p+1>p</math>, the sets {{mvar|X}} and {{mvar|Y}} must intersect.

The number {{mvar|u}} can be factored in Hurwitz quaternions:
<math display="block">1+l^2+m^2=(1+l\;\mathbf{i}+m\;\mathbf{j})(1-l\;\mathbf{i}-m\;\mathbf{j}).</math>

The norm on Hurwitz quaternions satisfies a form of the [[Euclidean domain|Euclidean]] property: for any quaternion <math> \alpha=a_0+a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k} </math> with rational coefficients we can choose a Hurwitz quaternion <math> \beta=b_0+b_1\mathbf{i}+b_2\mathbf{j}+b_3\mathbf{k} </math> so that <math> \mathrm{N}(\alpha-\beta)<1 </math> by first choosing <math> b_0 </math> so that <math> |a_0-b_0| \leq 1/4 </math> and then <math> b_1, b_2, b_3 </math> so that <math> |a_i-b_i| \leq 1/2 </math> for <math> i = 1,2,3</math>. Then we obtain
<math display="block">\begin{align}
\mathrm{N}(\alpha-\beta) & =(a_0-b_0)^2+(a_1-b_1)^2+(a_2-b_2)^2+(a_3-b_3)^2 \\
& \leq \left(\frac{1}{4} \right)^2+\left(\frac{1}{2}\right)^2+\left(\frac{1}{2}\right)^2+\left(\frac{1}{2}\right)^2=\frac{13}{16} < 1.
\end{align}</math>

It follows that for any Hurwitz quaternions <math>\alpha,\beta</math> with <math>\alpha \neq 0</math>, there exists a Hurwitz quaternion <math>\gamma</math> such that
<math display="block">\mathrm N(\beta-\alpha\gamma)<\mathrm N(\alpha).</math>

The ring {{mvar|H}} of Hurwitz quaternions is not commutative, hence it is not an actual Euclidean domain, and it does not have [[unique factorization domain|unique factorization]] in the usual sense. Nevertheless, the property above implies that every right [[ideal (ring theory)|ideal]] is [[principal ideal|principal]]. Thus, there is a Hurwitz quaternion <math>\alpha</math> such that
<math display="block">\alpha H = p H + (1-l\;\mathbf{i}-m\;\mathbf{j}) H.</math>

In particular, <math>p=\alpha\beta</math> for some Hurwitz quaternion <math>\beta</math>. If <math>\beta</math> were a unit, <math>1-l\;\mathbf{i}-m\;\mathbf{j}</math> would be a multiple of {{mvar|p}}, however this is impossible as <math>1/p-l/p\;\mathbf{i}-m/p\;\mathbf{j}</math> is not a Hurwitz quaternion for <math>p>2</math>. Similarly, if <math>\alpha</math> were a unit, we would have
<math display="block">(1+l\;\mathbf{i}+m\;\mathbf{j})H = (1+l\;\mathbf{i}+m\;\mathbf{j})p H+(1+l\;\mathbf{i}+m\;\mathbf{j})(1-l\;\mathbf{i}-m\;\mathbf{j})H \subseteq p H</math>
so {{mvar|p}} divides <math>1+l\;\mathbf{i}+m\;\mathbf{j}</math>, which again contradicts the fact that <math>1/p-l/p\;\mathbf{i}-m/p\;\mathbf{j}</math> is not a Hurwitz quaternion. Thus, {{mvar|p}} is not Hurwitz irreducible, as claimed.
}}

==Generalizations==
Lagrange's four-square theorem is a special case of the [[Fermat polygonal number theorem]] and [[Waring's problem]]. Another possible generalization is the following problem: Given [[natural number]]s <math>a,b,c,d</math>, can we solve

<math display="block">n=ax_1^2+bx_2^2+cx_3^2+dx_4^2</math>

for all positive integers {{mvar|n}} in integers <math>x_1,x_2,x_3,x_4</math>? The case <math>a=b=c=d=1</math> is answered in the positive by Lagrange's four-square theorem. The general solution was given by [[Ramanujan]].<ref>{{harvnb|Ramanujan|1916}}.</ref> He proved that if we assume, without loss of generality, that <math>a\leq b\leq c\leq d</math> then there are exactly 54 possible choices for <math>a,b,c,d</math> such that the problem is solvable in integers <math>x_1,x_2,x_3,x_4</math> for all {{mvar|n}}. (Ramanujan listed a 55th possibility <math>a=1,b=2,c=5,d=5</math>, but in this case the problem is not solvable if <math>n=15</math>.<ref>{{harvnb|Oh|2000}}.</ref>)

==Algorithms==

In 1986, [[Michael O. Rabin]] and [[Jeffrey Shallit]]<ref>{{harvnb|Rabin|Shallit|1986}}.</ref> proposed [[randomized algorithm|randomized]] [[polynomial-time algorithm]]s for computing a single representation <math>n=x_1^2+x_2^2+x_3^2+x_4^2</math> for a given integer {{mvar|n}}, in expected running time <math>\mathrm{O}(\log(n)^2)</math>. It was further improved to <math>\mathrm{O}(\log(n)^2 \log(\log(n))^{-1})</math> by Paul Pollack and Enrique Treviño in 2018.<ref>{{harvnb|Pollack|Treviño|2018}}.</ref>

==Number of representations==
{{main|Jacobi's four-square theorem}}
The number of representations of a natural number ''n'' as the sum of four squares of integers is denoted by ''r''<sub>4</sub>(''n''). [[Jacobi's four-square theorem]] states that this is eight times the sum of the [[divisor]]s of ''n'' if ''n'' is odd and 24 times the sum of the odd divisors of ''n'' if ''n'' is even (see [[divisor function]]), i.e.

<math display="block">r_4(n)=\begin{cases}8\sum\limits_{m\mid n}m&\text{if }n\text{ is odd}\\[12pt]
24\sum\limits_{\begin{smallmatrix} m|n \\ m\text{ odd} \end{smallmatrix}}m&\text{if }n\text{ is even}.
\end{cases}</math>

Equivalently, it is eight times the sum of all its divisors which are not divisible by 4, i.e.

<math display="block">r_4(n)=8\sum_{m\,:\, 4\nmid m\mid n}m.</math>

We may also write this as
<math display="block">r_4(n) = 8 \sigma(n) -32 \sigma(n/4) \ , </math>
where the second term is to be taken as zero if ''n'' is not divisible by 4. In particular, for a [[prime number]] ''p'' we have the explicit formula&nbsp;{{math|1=''r''<sub>4</sub>(''p'') = 8(''p'' + 1)}}.<ref name="Williams_2011">{{harvnb|Williams|2011|p=119}}.</ref>

Some values of ''r''<sub>4</sub>(''n'') occur infinitely often as {{math|1=''r''<sub>4</sub>(''n'') = ''r''<sub>4</sub>(2<sup>''m''</sup>''n'')}} whenever ''n'' is even. The values of ''r''<sub>4</sub>(''n'')/''n'' can be arbitrarily large: indeed, ''r''<sub>4</sub>(''n'')/''n'' is infinitely often larger than 8{{radic|log ''n''}}.<ref name="Williams_2011" />

==Uniqueness==
The sequence of positive integers which have only one representation as a sum of four squares of non-negative integers (up to order) is:

:1, 2, 3, 5, 6, 7, 8, 11, 14, 15, 23, 24, 32, 56, 96, 128, 224, 384, 512, 896 ... {{OEIS|A006431}}.

These integers consist of the seven odd numbers 1, 3, 5, 7, 11, 15, 23 and all numbers of the form <math>2(4^k),6(4^k)</math> or <math>14(4^k)</math>.

The sequence of positive integers which cannot be represented as a sum of four ''non-zero'' squares is:

:1, 2, 3, 5, 6, 8, 9, 11, 14, 17, 24, 29, 32, 41, 56, 96, 128, 224, 384, 512, 896 ... {{OEIS|A000534}}.

These integers consist of the eight odd numbers 1, 3, 5, 9, 11, 17, 29, 41 and all numbers of the form <math>2(4^k),6(4^k)</math> or <math>14(4^k)</math>.

==Further refinements==
Lagrange's four-square theorem can be refined in various ways. For example, [[Zhi-Wei Sun]]<ref>{{harvnb|Sun|2017}}.</ref> proved that each natural number can be written as a sum of four squares with some requirements on the choice of these four numbers.

One may also wonder whether it is necessary to use the entire set of square integers to write each natural as the sum of four squares. [[Eduard Wirsing]] proved that there exists a set of squares {{mvar|S}} with <math>|S| = O(n^{1/4}\log^{1/4} n)</math> such that every positive integer smaller than or equal to {{mvar|n}} can be written as a sum of at most 4 elements of  {{mvar|S}}.<ref>{{harvnb|Spencer|1996}}</ref>

==See also==
* [[Fermat's theorem on sums of two squares]]
* [[Fermat's polygonal number theorem]]
* [[Waring's problem]]
* [[Legendre's three-square theorem]]
* [[Sum of two squares theorem]]
* [[Sum of squares function]]
* [[15 and 290 theorems]]

==Notes==
{{reflist|refs=

<ref name=andrews>{{citation
 | last = Andrews | first = George E. | author-link = George E. Andrews
 | isbn = 0-486-68252-8
 | page = 144
 | publisher = Dover Publications
 | title = Number Theory
 | year = 1994}}</ref>}}

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 | location = Bangalore, India
 }}
*{{Cite book
 | last = Stillwell
 | first = John
 | author-link = John Stillwell
 | title = Elements of Number Theory
 | publisher = Springer
 | year = 2003
 | isbn = 978-0-387-95587-2
 | zbl = 1112.11002
 | doi = 10.1007/978-0-387-21735-2
 | series = Undergraduate Texts in Mathematics
 }}
*{{Cite journal
 | last = Sun
 | first = Z.-W.
 | author-link = Zhi-Wei Sun
 | title = Refining Lagrange's four-square theorem
 | journal = J. Number Theory
 | volume = 175
 | year = 2017
 | pages = 167–190
 | doi = 10.1016/j.jnt.2016.11.008
 | arxiv = 1604.06723
 | s2cid = 119597024
 }}
*{{Cite book
 | last = Williams
 | first = Kenneth S.
 | title = Number theory in the spirit of Liouville
 | series = London Mathematical Society Student Texts
 | volume = 76
 | publisher = [[Cambridge University Press]]
 | year = 2011
 | isbn = 978-0-521-17562-3
 | zbl = 1227.11002
 }}
*{{Cite book
 | last=Spencer
 | first=Joel
 | date=1996
 | pages=295–297
 | publisher=Springer US
 | language=en
 | doi=10.1007/978-1-4612-2418-1_22
 | isbn=9780387948263
| title=Number Theory: New York Seminar 1991–1995
 | chapter=Four Squares with Few Squares
 }}
*{{cite journal |first1=P. |last1=Pollack |first2=E. |last2=Treviño |title=Finding the four squares in Lagrange's theorem |journal=Integers |volume=18A |year=2018 |pages=A15 |url=http://math.colgate.edu/~integers/sjs15/sjs15.pdf}}

==External links==
*[http://planetmath.org/proofoflagrangesfoursquaretheorem Proof at PlanetMath.org]
*[http://www.alpertron.com.ar/4SQUARES.HTM Another proof]
*[http://www.alpertron.com.ar/FSQUARES.HTM An applet decomposing numbers as sums of four squares]
*[https://oeis.org/wiki/Index_to_OEIS:_Section_Su#ssq OEIS index to sequences related to sums of squares and sums of cubes]
*{{mathworld|urlname=LagrangesFour-SquareTheorem|title=Lagrange's Four-Square Theorem}}

{{Joseph-Louis Lagrange}}
{{authority control}}

{{DEFAULTSORT:Lagrange's Four-Square Theorem}}
[[Category:Additive number theory]]
[[Category:Articles containing proofs]]
[[Category:Squares in number theory]]
[[Category:Theorems in number theory]]