In mathematics, the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function. It states that
In other words,
The exponents 1, 2, 5, 7, 12, ... on the right hand side are given by the formula for k = 1, −1, 2, −2, 3, ...) and are called (generalized) pentagonal numbers. This holds as an identity of convergent power series for, and also as an identity of formal power series.
A striking feature of this formula is the amount of cancellation in the expansion of the product.
The identity implies a marvelous recurrence for calculating, the number of partitions of n:
or more formally,
where the summation is over all nonzero integers k (positive and negative) and is the kth pentagonal number.
Read more about Pentagonal Number Theorem: Bijective Proof, Partition Recurrence, Example Program
Famous quotes containing the words number and/or theorem:
“... is it not clear that to give to such women as desire it and can devote themselves to literary and scientific pursuits all the advantages enjoyed by men of the same class will lessen essentially the number of thoughtless, idle, vain and frivolous women and thus secure the [sic] society the services of those who now hang as dead weight?”
—Sarah M. Grimke (1792–1873)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)