|Date||Mar 24, 2023|
In 1997, Kaplansky conjectured that if two positive definite ternary quadratic forms with integer coefficients have perfectly identical integral representations, then they are isometric, both regular, or included either of two families of ternary quadratic forms. Recently, we proved the existence of pairs of ternary quadratic forms representing the same integers which are not contained in Kaplansky’s list. However, Do proved that Kapalansky's conjecture holds if only diagonal quadratic forms are considered. In this talk, we introduce Do's proof. Furthermore, we discuss about ternary sums of triangular numbers representing the same integers. This might be considered as a natural generalization of Do's result.