DATES:
Tuesday, November 16 (2010), at 1:55pm
PLACE: LIT 368
SPEAKER:
Duncan Buell (University of South Carolina)
ABSTRACT:
If a quadratic polynomial in x assumes square values for a sequence of
successive integers x, then those squares will have constant second difference
(equal to twice the lead coefficient).
Nontrivial sequences of four successive squares are known for quadratics
x2 + bx + c, but it is not known if sequences of five or more do/can exist.
We will present a complete
characterization of sequences of four successive squares that is unfortunately
not "effective" or "algorithmic".
Infinitely many sequences of eight successive squares have been shown to exist for
differences > 2.
Recently, Browkin and Brzezinski have shown that only finitely many "symmetric" sequences
exist with even length >10.
We will discuss these problems and relate them to simultaneous Pell equations,
elliptic curves, and real quadratic number fields.
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