Does this equation have positive integer solutions?

The only solution I can find for $a^2 + b^2 + c^2 = d^2 + e^2 + f^2$ is $a=0$, $b=0$, $c=0$, $d=0$, $e=0$, $f=0$. Are there any positive integer solutions? Any where none of $a,b,c,d,e,f$ are equal?

Solution 1:

Solving $a^2 + b^2 + c^2 = d^2 + e^2 + f^2$ has many many (boring) positive integer solutions.

Choose any bijection $\theta: \{a,b,c\} \to \{d,e,f\}$. Then, choose any positive integer values for $\{a,b,c\}$.

It must be true that $a^2 + b^2 +c^2 = \theta(a)^2 + \theta(b)^2 + \theta(c)^2 = d^2 + e^2 + f^2.$ As stated though, these are all pretty boring, and probably not quite what you're looking for. For less "boring" solutions, combining some known Pythagorean triples could help.

For example, $3^2 + 4^2 = 5^2$. Meanwhile $8^2 + 15^2 = 17^2$ (I think, somebody else can fact-check this triple). Thus,

$(3^2+4^2)+17^2 = 5^2 + (8^2+15^2)$

Solution 2:

You are asking, in a nontrivial vein, how many ways a positive integer can be represented as the sum of three squares.

It is widely known (related) which positive integers admit such representations. The question of how many sum of three squares representations exist for a given positive integer was explored by P. Bateman (1951) in a lengthy paper freely available from the AMS:

On the Representations of a Number as the Sum of Three Squares

and the classical literature on the subject (Dickson, Landau, Hardy, etc.) is well surveyed there up to the time.

In Dave Rusin's collection of posts related to this topic, we find him stating this interesting characterization:

In how many ways may one represent a given integer n as a sum of three squares? Eichler noted that for n squarefree, the number of ways is, as noted above, zero is n=7 mod 8. If n=3 mod 8, he showed the number of representations is 12h, where h is the class number of the field Q(sqrt(-n)). Otherwise, the number of representations is 24h.

Also Math.SE related: Representability as a Sum of Three Positive Squares or Non-negative Triangular Numbers.

A fast randomized algorithm for finding an integer solution $n = x^2 + y^2 + z^2$ was the topic of Rabin and Shallit algorithm. The discussion turns up an interesting connection to Hardy and Littlewoods's Conjecture H, about the number of ways to express $n = x^2 + p$ as the sum of a square and a prime.