Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars

Speaker:	Prof. Preda Mihailescu，University of Gottingen
Inviter:	魏达盛
Title:	Semilocal galois approximation for some popular Diophantine Equations
Language:	English
Time & Venue:	2023.08.16 13:30-14:30 南楼N913室
Abstract:	We consider the equation (x^p*+y^p)/(x+y) = p^e z^q, where x,y,z are non zero and mutually coprime integers, p and q are primes, not necessarily different and p > 3, while e = 0 is p does not divide z and 1 otherwise. We present an approach combining class field theory and approximation in semilocal products of completions at primes dividing x, which completely proves that the case p = q has no solutions, while providing good upper bounds for the case p neq q. We also indicate how to prove that the homogenous case of distinct exponents (Fermat-Catalan) has no solutions when either q > 2p or q does not divide the relative class number of the o-th cyclotomic field and that field contains no q-primary roots of unity which are not q-powers. We discuss further applications.