Prove that for any integer z that is a perfect square, there exists integers m and n such that m^2+n^2=zk.

Question

Let k be an integer such that there exist integers p and q with p^2+q^2=k.

 

Prove that for any integer z that is a perfect square, there exists integers m and n such that m^2+n^2=zk.

Answer 1

z = G² where G is some integer which be factored using the F.T.A. into a unique product of primes (this fact is not used in the proof).   

zk = z(p²+q² )= G²(p²+q²) = G²p²+G²q² 

 

Now we can assign m=Gp  and n=Gq  and get:

 n²+m² = G²p²+G²q² = zk  [done]

 

________

 

Example/sanity-check:   say z=25, p=2, q=7 :    k = p²+q² = 2²+7² = 53

 

zk = 25*53 = 1325

 

G=sqrt(z)=5     so  assign  m=5*2 = 10   and  n=5*7 = 35

 

  m²+n² =  10² + 35² = 100 + 1225 = 1325 = zk   (ok)