Does prayer become invalidated by the emission of something without sound or smell, and without certainty of it, and must it be repeated after a week has passed, and must the worshiper be informed of that?

The prayer of the questioner is valid, and he is not required to repeat it unless he is certain that something was emitted from him, because the default assumption is the validity of the prayer, and certainty is not removed except by a similar certainty. It is not a condition to hear a sound or smell an odor to be certain of emission; rather, the criterion is the attainment of certainty that ritual purity has been invalidated. If he becomes certain that something was emitted, he alone is obligated to repeat the prayer, and he is not obligated to inform those who prayed with him, because their prayer is valid as long as they were unaware of the invalidity of his prayer, just as Umar (may Allah be pleased with him) did.