Browsing by Subject "diophantine equations"

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    Solutions of diophantine equations as periodic points of p-adic algebraic functions, I
    (2016) Morton, Patrick; Department of Mathematical Sciences, School of Science
    Solutions of the quartic Fermat equation in ring class fields of odd conductor over quadratic fields K = Q(√-d) with -d Ξ 1 (mod 8) are shown to be periodic points of a fixed algebraic function T(z) defined on the punctured disk 0 < |z|2 ≥ 1/2 of the maximal unramified, algebraic extension K2 of the 2-adic field Q2. All ring class fields of odd conductor over imaginary quadratic fields in which the prime p = 2 splits are shown to be generated by complex periodic points of the algebraic function T, and conversely, all but two of the periodic points of T generate ring class fields over suitable imaginary quadratic fields. This gives a dynamical proof of a class number relation originally proved by Deuring. It is conjectured that a similar situation holds for an arbitrary prime p in place of p = 2, where the case p = 3 has been previously proved by the author, and the case p = 5 will be handled in Part II.
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    Solutions of diophantine equations as periodic points of p-adic algebraic functions, II: The Rogers-Ramanujan continued fraction
    (2019) Morton, Patrick; Mathematical Sciences, School of Science
    In this part we show that the diophantine equation X5+Y5=ε5(1−X5Y5) , where ε=−1+5√2 , has solutions in specific abelian extensions of quadratic fields K=Q(−d−−−√) in which −d≡±1 (mod 5 ). The coordinates of these solutions are values of the Rogers-Ramanujan continued fraction r(τ) , and are shown to be periodic points of an algebraic function.