If a number 'c' is a sum of 2 squares of say 'x' and 'y',can '2c' be also expressed as a sum of 2 different squares?
The answer is yes.
You have to work out (x+y) and (x-y) ,find their squares and add up.
We already have c=x^2+y^2
We now have (x+y)^2+(x-y)^2= (x^2+2xy+y^2)+(x^2-2xy+y^2)=2(x^2+y^2)=2c
Example-Let x=17 and y=9.So c=289+81=370
We also have (x+y)=26 and (x-y)=8.Their squares are 676 and 64 giving a total of 740 which is the same as '2c'.
As a corollary if we have three numbers x,y and z and their squares add upto 'c',we can express '4c' as the sum of 6 squares viz of x+y..x-y...y+z...y-z....z+x and z-x.
The answer is yes.
You have to work out (x+y) and (x-y) ,find their squares and add up.
We already have c=x^2+y^2
We now have (x+y)^2+(x-y)^2= (x^2+2xy+y^2)+(x^2-2xy+y^2)=2(x^2+y^2)=2c
Example-Let x=17 and y=9.So c=289+81=370
We also have (x+y)=26 and (x-y)=8.Their squares are 676 and 64 giving a total of 740 which is the same as '2c'.
As a corollary if we have three numbers x,y and z and their squares add upto 'c',we can express '4c' as the sum of 6 squares viz of x+y..x-y...y+z...y-z....z+x and z-x.
