Square Curious Numbers
Neelima Borade and Jacob Mayle
University of Illinois at Chicago
Department of Mathematics, Statistics and Computer Science
Chicago, IL 60302-7045
A curious number is a palindromic number whose base-ten representation has
the form a ··· ab ···
ba ··· a. In this paper, we determine
all curious numbers that are perfect squares. Our proof involves reducing
the search for such numbers to several single variable families. From
here, we complete the proof in two different ways. The first approach is
elementary, though somewhat ad hoc. The second entails studying integral
points on elliptic curves and is more systematic. Following these proofs,
we consider curious numbers with respect to other bases. We use the
elliptic curve technique to determine all perfect squares among the
curious numbers in bases 2, 4, 6, and 12.
Full version: pdf,
(Concerned with sequences
Received October 11 2020; revised versions received June 19 2021; June 20 2021.
Published in Journal of Integer Sequences,
June 29 2021.
Journal of Integer Sequences home page