Four square Lagrange Theorem.

Question

Today teacher tell us that every natural number is the sum of four positive squares.

and let us this work:

write $747004$ as the sum of $4$ squares.

I'm at high school and don't understand the proof in Wikipedia.

Answer 1

I'm lucky, if you rest $1$ to $747004$ you have a number wich is divisible by $3$ and $1$ is a square

$$747004=1^2+499^2+499^2+499^2$$

Since Geoff Robinson comment, we have: $$ 747004=444^2+606^2+414^2+106^2 $$

Answer 2

there are 12537 integer quadruples $w \geq x \geq y \geq z \geq 0$ with $w^2 + x^2 + y^2 + z^2 = 747004.$ Since $747004 = 4 \cdot 186751,$ and $186751 \equiv 7 \pmod 8.$ As a result, $747004$ is not the sum of three squares, so there are no quadruples with any zeroes. Here are some form the beginning, middle, and end:

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     863      47       5       1
     863      43      19       5
     863      41      23       5
     863      37      29       5
     863      35      31       7
     863      35      29      13
     862      62      10       4
     862      60      18       6
     862      58      20      14
     862      54      30      12
     862      52      34      10
     862      50      38       4
     862      50      28      26
     862      46      38      20
     862      42      36      30
     861      75       7       3
     861      69      29       9
     861      65      27      27
     861      63      33      25
     861      61      39      21
     861      57      47      15
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     499     499     499       1
     499     499     491      89
     499     499     461     191
     499     499     419     271
     499     497     437     245
     499     481     421     299
     499     475     467     233
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     462     438     430     396
     461     443     425     397
     461     437     433     395
     460     446     422     398
     458     452     410     406
     457     455     433     379
     457     453     411     405
     457     441     435     393
     455     449     443     377
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=