Symmetric functions on three parameters being perfect squares - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T12:27:59Z http://mathoverflow.net/feeds/question/94310 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/94310/symmetric-functions-on-three-parameters-being-perfect-squares Symmetric functions on three parameters being perfect squares Hej 2012-04-17T17:56:30Z 2012-04-18T02:18:52Z <p>Is it possible for $x+y+z, xy+yz+zx$, and $xyz$ to be perfect squares at the same time for positive integer values of $x,y,z$?</p> http://mathoverflow.net/questions/94310/symmetric-functions-on-three-parameters-being-perfect-squares/94346#94346 Answer by Noam D. Elkies for Symmetric functions on three parameters being perfect squares Noam D. Elkies 2012-04-18T02:16:24Z 2012-04-18T02:16:24Z <p>Yes. By straightforward search the smallest example is $\lbrace x,y,z \rbrace = \lbrace 45,64,180 \rbrace$, with $$(t+45) (t+64) (t+180) = t^3 + 17^2 t^2 + 150^2 t + 720^2.$$ Given any solution $(x,y,z)$ we may produce infinitely many others (other than the trivial scaling $(c^2 x, c^2 y, c^2 z)$) by using the theory of elliptic curves to find <em>rational</em> $z'$ such that $x+y+z'$, $xy+yz'+z'x$, and $xyz'$ are all squares, at which point $(d^2 x, d^2 y, d^2 z')$ works for any integer $d>0$ such that $d^2 z' \in {\bf Z}$. For example, in $\lbrace 45,64,180 \rbrace$ we may replace $64$ by $(460163992/28591599)^2$, and then multiply through by $28591599^2$ to obtain the new solution $$\lbrace<br> 28591599^2 \cdot 45, \phantom{+} 460163992^2, \phantom{+} 28591599^2 \cdot 180 \rbrace .$$</p> <p>A complete parametrization is not possible, because it would be tantamount to a rational parametrization of the surface $$S: xy + yz + zx = r^2, \phantom{and} (x+y+z)xyz = s^2$$ in projective $(1:1:1:1:2)$ space, and that surface is K3. If I did this right, $S$ is a "singular" K3 surface, i.e. has Picard number $20$ which is maximal for a K3 surface in characteristic zero, and the N&eacute;ron-Severi group ${\rm NS}(S)$ has rank $20$ and discriminant $-48$, and consists of (classes of) divisors defined over ${\bf Q}(i)$.</p> <p>It is actually quite common for natural Diophantine equations to give rise to K3 surfaces of maximal or nearly-maximal Picard number, but that's a story for another time.</p>