Nicomachus of Gerasa is mentioned in a small number of sources and we can date him fairly accurately. From his writings on numbers and music, it is clear that Nicomachus was a Pythagorean. Porphyry says that he was one of the leading members of the Pythagorean School.

Nicomachus wrote *Introduction to Arithmetic*, which was the first work to treat arithmetic as a separate topic from geometry. Unlike
Euclid, Nicomachus gave no abstract proofs of his theorems, merely stating theorems and illustrating them with numerical examples. The work contains elementary errors which indicate that he did not have proofs to give.

An example of this, consider the results which Nicomachus quotes on perfect numbers. He states that the n^{th} perfect number has n digits, and that
all perfect numbers end in 6 and 8 alternately. These are false, but are reasonable guesses based on the four perfect numbers known during his time, namely 6, 28, 496 and 8128.

The work contains the first multiplication table in a Greek text. It is also remarkable in that it contains Arabic numerals, not Greek ones. For over 1000 years *Introduction to Arithmetic* was the standard arithmetic text, despite the many errors and superstitous numerology. Most Arabic texts on number theory written by mathematicians were influenced by both Euclid and Nicomachus.

Nicomachus also wrote two volumes of *The Theology of Numbers*, which was completely concerned with mystic properties of numbers.
Another work by Nicomachus which has survived is *Manual of Harmonics*, which is a work on musical notes and the octave. Both Porphyry and Iamblichus wrote biographies of Pythagoras which quote from Nicomachus. From this evidence some historians have conjectured that
Nicomachus also wrote a biography of Pythagoras and, although there is no direct evidence, it is indeed quite possible.