MathDB
infimum = 0

Source: Romanian TST 1978, Day 3, P6

September 30, 2018
algebra

Problem Statement

a) Prove that 0=inf{x2+y3+y5x,y,zZ,x2+y2+z2>0} 0=\inf\{ |x\sqrt 2+y\sqrt 3+y\sqrt 5|\big| x,y,z\in\mathbb{Z} ,x^2+y^2+z^2>0 \}
b) Prove that there exist three positive rational numbers a,b,c a,b,c such that the expression E(x,y,z):=xa+yb+zc E(x,y,z):=xa+yb+zc vanishes for infinitely many integer triples (x,y,z), (x,y,z), but it doesn´t get arbitrarily close to 0. 0.