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Let $m,n$ be natural numbers and $m\not=n$. Show that $m^4+4n^4$ can be expressed as the sum of four squares of natural numbers.

That is, express $m^4+4n^4$ as $A^2+B^2+C^2+D^2$ where $A,B,C,D$ is an expression in $n$ or/both $m$.

@Tim: Yes, you are completely right. This is the most careless mistake ever!! @Robert: And yes I want the sum of four squares of expressions (not necessarily polynomials, can contain the term mn) in m and/or n with integer coefficients.

And what is the significance of assuming $m\not=n?$

Any help will be greatly appreciated. Many many thanks!

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  • $\begingroup$ Any positive integer can be expressed as the sum of four squares (Lagrange's four-square theorem). But apparently you want something else: $m^4 + 4 n^4$ as the sum of four squares of polynomials in $m$ and $n$ with integer coefficients? $\endgroup$
    – Robert Israel
    Commented Dec 22, 2013 at 8:02
  • $\begingroup$ Your first question can be answered by saying that you made an error. You tried to apply the sum of cubes formula to a sum of squares! $\endgroup$
    – Tim
    Commented Dec 22, 2013 at 8:04
  • $\begingroup$ @Tim: You are right, thanks. I don't what's in my mind... I have edited it. $\endgroup$
    – user71346
    Commented Dec 22, 2013 at 8:17
  • $\begingroup$ $(m-m)^2+(n-n)^2+(m^2)^2+(2n^2)^2.$ $\endgroup$
    – John Bentin
    Commented Dec 22, 2013 at 8:46
  • $\begingroup$ Thanks John! Your answer is pretty striking, it didn't come up in my mind in the first place. But I guess it will work if we assume 0 as a natural number. $\endgroup$
    – user71346
    Commented Dec 22, 2013 at 9:01

1 Answer 1

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$$m^4+4n^4=(m^2)^2+(2n^2)^2=(m^2+2n^2)^2-(2mn)^2$$ $$=(m^2+2n^2-2mn)(m^2+2n^2+2mn)=\{(m-n)^2+n^2\}\{(m+n)^2+n^2\}$$

Now multiply

As $m\ne n, (m-n)^2>0$

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  • $\begingroup$ Thanks! This should not be a difficult question. I should be more careful. $\endgroup$
    – user71346
    Commented Dec 22, 2013 at 8:24
  • $\begingroup$ @user71346, my pleasure. Yes, nice to hear that "not difficult" $\endgroup$
    – lab bhattacharjee
    Commented Dec 22, 2013 at 8:26

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