Question

Hi all. I am dealing with e problem and I hope you can help me. I have already proved this:

Let us suppose that integers m and n can be written as sum of squares of two integers. Prove that m*n can also be written as sum of squares of two integers.

Now I am trying to prove that m*n can also be written as sum of squares of four integers.

Can you help me on this?
Thank You for reading my post.

Answer 1

let m=a^2+b^2
let n=x^2+y^2

mn = (a^2+b^2)(x^2+y^2)

Answer 2

mn = (ax)^2 + (ay)^2 + (bx)^2 + (by)^2

Answer 3

how did you prove the first one?

Answer 4

a^2+b^2=m
c^2+d^2=n

(ac)^2+(ad)^2+(bc)^2+(bd)^2=mn
(ac)^2+2acbd+(bd)^2-2acbd+(ad)^2-2adbc+(bc)^2+2adbc=mn

(ac+bd)^2+(ad-bc)^2=mn

That's how i proved it:)

Answer 5

cool, so in your proofing of the first one, youve actually proven the second one

Answer 6

But the problem is that m should be equal to a^2+b^2+c^2+d^2 and n should we equal to f^2+l^2+g^2+s^2.

Answer 7

that is not what is stated originally

Suppose: m and n can be written as sum of squares of two integers

m = (a^2 + b^2) is the sum of two integers square
n = (x^2 + y^2) is the sum of two integers square

Prove that: m*n can be written as sum of squares of four integers.

if we let m*n = k, such that \(k = s^2+t^2+u^2+v^2\)

Answer 8

the inital conditions for m and n havent changed as far as I am aware

Answer 9

Oh it's my mistake,I thought i have mentioned that :S

Answer 10

so to be clear:

are we saying that the inital conditions for m and n have changed?

or have they remained the same and youve simply misread the problem?

Answer 11

Hi all. I am dealing with a problem and I hope you can help me. I have already proved this:

Let us suppose that integers m and n can be written as sum of squares of two integers. Prove that m*n can also be written as sum of squares of two integers.


Now I am trying to prove this:Let us suppose that integers m and n can be written as sum of squares of four integers. Prove that m*n can also be written as sum of squares of four integers.

Answer 12

a+b+c+d
w+x+y+z
----------
aw+bw+cw+dw
ax+bx+cx+dx
ay+by+cy+dy
az+bz+cz+dz

looks to me like you just have to work out 4 completed squares

Answer 13

a+b
x+y
-----
ax+bx+2abxy
ay+by -2axby


a+b+c+d
w+x+y+z
----------
aw+bw +2abwx +cw+dw +2cdwx
ax+bx -2abwx +cx+dx -2cdwx
ay+by +2abyz +cy+dy +2cdyz
az+bz -2abyz +cz+dz -2cdyz

Answer 14

i spose that 8 integers :)

Answer 15

at least thats better than 16 integers that we started with

rename the 8 integers in terms that can be reduced to 4 terms is an idea