Recurrence Relations and the Algebraic Irrationality of Ζ(3)

We apply the theory of disconjugate linear recurrence relations to the study of irrational quantities in number theory. In particular, we show that there exists a four-term real linear recurrence relation whose solutions allow us to produce a tractable equivalent criterion for the quadratic irrationality of ζ(3), where ζ is the classic Zeta function of Riemann or, for that matter, the quadratic irrationality of any other irrational that arises either as a result of an Apéry-like argument, or from a continued fraction expansion. The result is amplified to produce a criterion for the transcendence of such irrational numbers.