Question

4. Show that the equation x + y + 32 - 2xyz has no nontrivial integer solutions.

Answer 1

show that eq x² + y² + 2² = 2xyz has no non trivial integer soling. sor Jwe use two hemma's Lemma l;- ff x² + y² + ² = 2n xyz for some In y 2 t 2, ni at thin x, y, z are all even Reason. x, y, 2 cannot all be odd because the LILS would then be old and RHS even) so whou let - 2x so can becomes a + y² + 2 ² = 2 ntx y z to Чly t? — м both &lta, Umma 2! If x² + y² + 2² = 2ny z for some nezt ng2t2 = 2xy = 2y , Z = 22, then 1² + y² + 2,2- ehtny z, Hemma I and remmar together imply that in 24 y² + 7² - 20nyz for tome nt zt, y ze 2 then , you are dissible by assitrarily large powers of 2 Tier kt at we have 2k|x, y, z which I can only be possible if. ayalee - which is a trivial suo # L x2 + y2 + z² = 2x y z has only trivial sol. There is no non trivial integer sur inned with ImScanner