In the fifth week of pregnancy the embryo has already been implanted in the uterus. At this point, the embryo begins to secrete the hormone beta-hCG, known as the pregnancy hormone, which is the one detected in a pregnancy test. Therefore, if the woman is really pregnant, the test will be positive in the fifth week. However, there can always be failures and therefore it is recommended to repeat the test after a few days in case of doubt.
You can read more about this in the following link: Pregnancy Symptoms but Negative Test, What Else Could It Be?
Suggested for you: Five Weeks Pregnant: Development of the Baby and Symptoms of Mother.