Has nontrivial solution in positive integers of a diophantine equation as follows ?
$$x_1^2+x_2^2+x_3^2+x_4^2=y_1^2+y_2^2+y_3^2+y_4^2$$
Where trivial solutions are $x_i=y_j$.
Can you send me any reference for this question?
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4$\begingroup$ A general solution can be obtained simply by choosing different sets of signs in a quaternion norm identity, analogous to how the general solution of the sum of two squares equation of the same type can be obtained by considering norms of Gaussian integers. $\endgroup$– John R RamsdenCommented Jul 5, 2017 at 13:28
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2$\begingroup$ Or you can use that $x_1^2-y_1^2=N$ has non-trivial solutions in integers iff $N$ is not 2 mod 4. $\endgroup$– R.P.Commented Jul 5, 2017 at 15:21
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$\begingroup$ See OEIS sequence A024367 $\endgroup$– Robert IsraelCommented Jul 5, 2017 at 19:05
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2 Answers
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There are many non-trivial solutions. By Lagrange's four-square theorem, every natural number can be represented as a sum of $4$ squares (with $0^2$ allowed), and Jacobi's four-square theorem gives the number $r_4(n)$ of representations of $n$. Note that $r_4(n)$ double counts solutions, but since $r_4(n)$ is arbitrarily large infinitely often, this clearly implies that there exist infinitely many $n$ with two distinct representations (without $0^2$).
answered Jul 5, 2017 at 13:41
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Besides Jacobi's four square theorem, you can also show that there are nontrivial solutions because there are $k^4$ quadruples of integers from $0$ through $k-1$, and all of the sums of squares are at most $4k^2$ which is $o(k^4)$. So, the average number of ways to write a number up to $n$ as a sum of four squares goes to infinity as $n \to \infty$.
In Mathematica, you can ask for distinct ways to express an integer as a sum of four squares as follows:
PowersRepresentations[100,4,2]
{{0, 0, 0, 10}, {0, 0, 6, 8}, {1, 1, 7, 7}, {1, 3, 3, 9}, {1, 5, 5, 7}, {2, 4, 4, 8}, {5, 5, 5, 5}}
Length[PowersRepresentations[10000, 4, 2]]
67
answered Jul 5, 2017 at 13:47
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$\begingroup$ Thank to You very much, there is geometry of this problem: mathoverflow.net/questions/257693/… $\endgroup$ Commented Jul 5, 2017 at 13:57