A system of equations having no nontrivial solutions

Abstract

L The object of this note is to prove the THEOREM: The system of equations af + a~ + . + a~_1 = br + b~ + . . . + b~_l' r=2,3, .. . , n ; (1) has no nontrivial solutions in positive integers. In what follows, we write Ar for a~+ a;+ "'" 1; Br for bi+ b;+ + b~_ I' r"'" 1; and all small letters denote integers"'" 0 unless stated otherwise. 2. PROOF OF THE THEOREM: Let at, a2, ... , a,,_1 From (4) we have AI 1 0 0 0 A2 AI 2 0 0 A3 A2 AI 3 0 r!'\r =