Any D(4)-quintuple contains a regular quadruple
by
Alan Filipin
University of Zagreb

A set of m positive integers is called a D(4)-m-tuple, if the product of any two of its distinct elements increased by 4 is a perfect square. Moreover, we call a D(4)-quadruple {a, b, c, d} such that d > max{a, b, c} regular if d=a+b+c+[1/2](abc+rst), where r, s and t are positive integers given by ab+4=r², ac+4=s², bc+4=t². There is a conjecture that all D(4)-quadruples are regular, which would imply that there does not exist a D(4)-quintuple. It is proven that there is no D(4)-sextuple. In this talk we prove that any D(4)-quintuple contains a regular quadruple, i.e. that we cannot extend an irregular D(4)-quadruple with a larger element.

Date received: March 11, 2008