[FOM] Another contructivist query

[Bill Taylor]

I recall reading something to the effect that if a number theoretic
statement has been proved classically, and it states something that
(in principle) needs only a finite number of cases to check,
then there also exists a constructive proof.

This would mean, for example, that there is a constructive proof,
that there exists a prime between  10^10^10^10 and 2x10^10^10^10.

(a) Is the above correct?

(b) Would a full-blown subjectivist intuitionist assent to the above?