A Family of Irreducible Polynomials of Degrees 6 to 62 and Corresponding Salem Numbers

AbstractThis article presents a family of irreducible, palindromic polynomials with integer coefficients,ranging in degree from 6 to 62. For each polynomial, the associated Salemnumber – a real algebraic integer greater than 1 whose remaining conjugates lie on orinside the unit circle – is computed. The polynomials satisfy the defining criteria of Salempolynomials: irreducibility over Z, palindromicity, exactly one root outside the unit circle,and all remaining roots on the unit circle. The computations were performed using Sage-Math, with algorithmic assistance from DeepSeek. The discovery stems from an originalapproach by the author.