Difference of two sums of three squares

Question

I have proved that every integer is the difference of two sums of three squares, i.e.,

$n = (a^2 + b^2 + c^2) - (d^2 + e^2 + f^2)$

Is this result publishable?

Answer 1

You might enjoy the 1939 book by Leonard Eugene Dickson, called Modern Elementary Theory of Numbers. One of the things he does is find all quadratic forms in three variables that are "universal." One familiar example is $x^2 + y^2 - z^2.$

Answer 2

Any integer not congruent to $2$ modulo $4$ can be written as the difference of two integer squares. Hence the sum of two differences of two integer squares can always be made to represent any number. For example, let $n \ge 1$ be the number, and let $a \ge 1$ and $b \ge 1$ be any two numbers such that $a,b \not\equiv 2\!\pmod{4}$ and $a+b=n$. Then $a=a_1^2-a_2^2$ and $b=b_1^2-b_2^2$, and $$n = a + b = (a_1^2-a_2^2)+(b_1^2-b_2^2) = (a_1^2+b_1^2)-(a_2^2+b_2^2).$$ This can be extended to any number of squares by choosing $n=a+b+c$ or $n=a+b+c+d$, and so on; your example is $n=a+b+c$.

So the answer to your question is basically "no"… although it might be a worthwhile paper if you showed the generalization(s), and found some interesting applications of the results.