A NEW APPROACH TO THE STUDY OF $D(-1)-$QUADRUPLES (Analytic Number Theory and Related Areas)

Abstract

Problems found in Diophantus's books on Arithmetics prompted intense research activity in recent years. The modern treatment evolved from the unifying notion of D(n)-m-sets. After a short overview of the main questions in this area; we concentrate on D(-1)-sets. It is conjectured that no D(-1)-quadrupleb exist, but the issue is still open. In the literature one finds several properties a hypothetical D(-1)-quadruple necessarily has. After sketching the strategy producing such results, a novel approach is pointed out. This survey is based on work in progress performed jointly with N. C. Bonciocat (Bucharest, Romania) and M. Mignote (Strasbourg, France).