Generalized Erdös and Obláth theorem for polynomialfactorial Diophantine equations
 Date
 October 23 (Tue) at 10:30  11:30, 2018 (JST)
 Speaker

 Mr. Wataru Takeda (Nagoya University)
 Language
 English
Diophantine equations are equations where only integer solutions are accepted. There are many types of Diophantine equations and many results are known. Our Diophantine equation is of the form x^n+y^n=m!. Erdös and Obláth showed that the Diophantine equation x^2+y^2=m! has only two positive integer solutions (x,y,m)=(1,1,2),(12,24,6).
In this talk, the factorial function m! is replaced with a generalized factorial function Π(m) over number fields. Then whether there are infinitely many solutions or not depends on the number field. We give necessary and sufficient condition for existence of infinitely many solutions of x^2+y^2=Π(m). More generally, we introduce an observation for higher degree equation x^n+y^n=Π(m).