Editing Lagrange's four-square theorem (section)

=== 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''.
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