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Given integers b,n, what are integer C,$a_i$ who solves $a_1^2+a_2^2+....+a_n^2 = b.C^2$ ??

Example for n=4 , b=7-> $a_1^2+a_2^2+a_3^2+a_4^2 = 7C^2$

or

for n=3, b=1 -> $a_1^2+a_2^2+a_3^2 = C^2$

Please if any reference, book, author I will appreciate thanks

Miguel Velilla
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1 Answers1

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Every positive integer is the sum of four squares, in particular there exist $a_1,a_2,a_3,a_4$ with $a_1^2+\cdots +a_4^2=7C^2$ for every $C$. For two squares and three squares there are well-known theorems as well:

Sum of one, two, and three squares

Show that an integer of the form $8k + 7$ cannot be written as the sum of three squares.

Dietrich Burde
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  • Currently I know all b are in the form b = $x_1^2+x_2^2..+..x_n^2$ and until I notice or there infinite solutions for every case or no solutions.at all. I don´t know if last statement is true, that´s to say exists some solution for b not being in the form of sum of squares. – Miguel Velilla Sep 23 '18 at 16:24
  • I mean for example to solve $x_1^2+x_2^2=7.c^2$ , since 7 can not be sum of two squares then I suppose there is no solution for this case. – Miguel Velilla Sep 23 '18 at 16:27
  • This can be decided by Legendre's theorem, see here. – Dietrich Burde Sep 23 '18 at 16:48
  • It shows that $x_1^2+x_2^2=7c^2$ has no integer solution. We can also see this directly, see here. – Dietrich Burde Sep 23 '18 at 19:02
  • I got to solve now more complete forms t1a1^2+...+tnan^2=b.c^2 by using parametric forms

    Example for n=5, b=6 this sequence $1.a_1^2+2.a_2^2+3.a_3^2+4.a_4^2+5.a_5^2 = 6.C^2$

    one solution (among infinite other) is $1.77^2+2.61^2+3.67^2+4.(2)^2+5.(4)^2=6.67^2$ I am writing development for this method soon will post here

     – Miguel Velilla Sep 23 '18 at 21:36
  • For previous example .... = 7.C^2 $47^2+38^2+37^2+75^2=7.39^2$ – Miguel Velilla Sep 23 '18 at 21:41