An increase in serum cortisol following the administration of ACTH is indicative of normal functioning adrenal glands. This contradicts the diagnosis of primary adrenal insufficiency.

In primary adrenal insufficiency, the adrenal glands are damaged and cannot respond to ACTH stimulation to produce cortisol. Therefore, in a rapid ACTH stimulation test, a patient with primary adrenal insufficiency will show decreased or unchanged serum cortisol levels, despite the administration of synthetic ACTH in the test.

The increase in urinary 17-hydroxycorticosteroids suggests increased adrenal gland activity, which is seen in conditions like Cushing's syndrome (excessive cortisol production) not primary adrenal insufficiency.

Low urinary 17-hydroxycorticosteroids may correlate with reduced adrenal gland activity, but this result alone isn't sufficient to diagnose primary adrenal insufficiency because it doesn't identify the source of the problem.