Journal of Integer Sequences, Vol. 24 (2021), Article 21.7.4

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.

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(Concerned with sequences A002113 A010785 A335779.)

Received October 11 2020; revised versions received June 19 2021; June 20 2021. Published in Journal of Integer Sequences, June 29 2021.

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