Nobody cares about the halting problem from a specific standpoint. If it's algebraically guaranteed to halt, it halts. When it halts is entirely meaningless for purposes of the halting problem.

But the thing is, it isn't algebraically guaranteed either - we still don't know if an odd perfect number exists. (I'm not sure where the "Definitely. Somewhere in 10^300 range." is coming from. There's nothing definite about the existence, only constraints that need to be satisfied if it exists)