On the binary digits of n and n^2

Let s(n) denote the sum of digits in the binary expansion of the integer n. Hare, Laishram and Stoll (2011) studied the number of odd integers such that s(n) = s(n^2) = k, for a given positive integer k. The remaining cases that could not be treated by theses authors were k = 9, 10, 11, 14 or 15. In this talk, I will present the results of our article on the cases k = 9, 10 and 11 and the difficulties to settle for the two remaining cases k = 14 and 15.

A related problem is to study perfect squares of odd integers with four binary digits. Bennett, Bugeaud and Mignotte (2012) proved that there are only finitely many solutions and conjectured that the set of solutions is composed of 13, 15, 47 and 111. In the same paper, we give an algorithm to find all solutions with fixed sum of digits value, supporting this conjecture, as well as show related results for perfect squares of odd integers with five binary digits.

This is joint work with Aloui, Jamet, Kaneko, Kopecki and Stoll.