Constructivists reject the notion of an integer whose value is not known (more precisely, whose value cannot be computed in finite time) so there is no room to

Constructivists reject the notion of an integer whose value is not known (more precisely, whose value cannot be computed in finite time) so there is no room to even ask of "it" is odd. Infinite vs finite is an important distinction. A fact that takes infinite time to prove is not constructive and is likewise not computable.