Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.5

On Polynomial Pairs of Integers

Martianus Frederic Ezerman
Division of Mathematical Sciences
School of Physical and Mathematical Sciences
Nanyang Technological University
21 Nanyang Link
Singapore 637371

Bertrand Meyer and Patrick Solé
Telecom ParisTech
46 rue Barrault
75634 Paris Cedex 13


The reversal of a positive integer A is the number obtained by reading A backwards in its decimal representation. A pair (A, B) of positive integers is said to be palindromic if the reversal of the product A × B is equal to the product of the reversals of A and B. A pair (A, B) of positive integers is said to be polynomial if the product A × B can be performed without carry.

In this paper, we use polynomial pairs in constructing and in studying the properties of palindromic pairs. It is shown that polynomial pairs are always palindromic. It is further conjectured that, provided that neither A nor B is itself a palindrome, all palindromic pairs are polynomial. A connection is made with classical topics in recreational mathematics such as reversal multiplication, palindromic squares, and repunits.

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(Concerned with sequences A002778 A002779 A004023 A062936 A156317.)

Received October 29 2012; revised version received August 5 2014; February 14 2015. Published in Journal of Integer Sequences, February 14 2015.

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