Pythagoras is one of the most famous names in the history of thought and one of the least recoverable people in it. He wrote nothing. No text by him was known even in antiquity. The earliest references to him, within a century or so of his death, are brief and largely hostile: Heraclitus calls him a practitioner of inquiry who made his own wisdom out of other people's writings; Xenophanes tells a story about him recognising the soul of a dead friend in a beaten puppy, which is mockery, but which is also the earliest evidence that he taught the transmigration of souls; Herodotus mentions him in passing. The detailed accounts (the golden thigh, the appearance in two cities at once, the taming of a bear, the catalogue of dietary rules), come from biographies written seven or eight centuries later by Porphyry and Iamblichus, who were Neoplatonists with a case to make.
Between those two bodies of evidence sits the problem. The later material is abundant, vivid and almost entirely untestable; the early material is meagre but contemporary. Modern scholarship, following Walter Burkert's study of 1962, has generally taken the early evidence as controlling, and the resulting picture is of a charismatic religious authority rather than a scientist: a man known for teaching what happens to the soul after death, for founding a community with a rule of life, and for a personal authority so complete that his followers were said to settle arguments by saying that he himself had said so.
Two things can be attributed to him with reasonable confidence. He taught metempsychosis, the passage of the soul into other bodies including animal ones, which is what Xenophanes is mocking. And he founded a society at Croton that lived by a shared discipline and became politically significant enough to be violently suppressed.
Almost everything else is contested, including the item everyone knows. The relation between the sides of a right triangle was in use in Mesopotamia a thousand years before him, there is no early evidence connecting him to a proof, and the attribution appears only in late sources. The famous prohibition on beans is reported in antiquity, but it is reported alongside mutually exclusive explanations, which is usually a sign that the reason had been forgotten while the rule survived.