Square values of Littlewood polynomials

We study the square values of Littlewood polynomials. Using various methods we give all these values for the degrees \(n=3, 5\) $n=3,5$ and \(n\le 24\) $n\le 24$ even. Beside this, we gather computational data (by providing all solutions in a certain range) for n odd with \(n\le 17\) $n\le 17$ . We propose some striking problems for further research, as well.