Pythagoras wrote nothing, or nothing that survived. Every statement about what he taught passes through followers, rivals, and admirers writing centuries later, and the tradition grew rather than shrank with distance, the fullest lives of him, by Porphyry and Iamblichus, were written some eight hundred years after his death, by Neoplatonists with a philosophical interest in making him a divine sage.
Walter Burkert's study of the sources separated two Pythagorases who had long been treated as one. He proceeded by weighing everything against Plato, Aristotle and their immediate pupils, and treating the Neoplatonic lives as back-projection. What survives that test is a religious and political figure: the head of a community at Croton with initiatory rules, dietary prohibitions, wonder-working, and a doctrine of the transmigration of souls. The mathematician and cosmologist appears in the record substantially later. Philolaus, in the fifth century, is the earliest reliable source for Pythagorean doctrine; Archytas is the earliest Pythagorean who is demonstrably a working mathematician.
This is a majority position rather than a settled one. Leonid Zhmud has argued at length for a more scientific early Pythagoreanism and against the distinction Burkert relies on. The point that has held against the challenge is narrow and decisive: no early source ascribes the mathematics to Pythagoras himself.
So the honest position on the Pythagorean theorem is that the relation was known to Babylonian scribes more than a millennium before him, it is on the tablet catalogued as Plimpton 322, and stated geometrically in the Indian Sulba Sutras, and that no early evidence connects either the discovery or a proof to Pythagoras personally. The attribution traces to Proclus, writing roughly a thousand years later, and even he credits only the recognition of the truth rather than its demonstration.
What is not in doubt is the consequence. Whoever first said that number is the principle of things, the idea entered Western thought through this school, and it runs from Plato's Timaeus through Renaissance harmonics to Kepler and into the numerology and sacred geometry of the modern esoteric tradition. The historical thinness of the man is a poor guide to the size of his influence.