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Izzhov
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I am aware that there are several generator functions for the Partition Function p(n), but does anyone know if a closed form formula exists for this function?
Izzhov said:"[A closed-form formula is] a single arithmetic formula obtained to simplify an infinite sum in a general formula." -Wikipedia
A closed-form formula is when you take a formula with an infinite sum, such asCRGreathouse said:That doesn't tell me what it is at all. Further, I don't see what closed form formulas have to do with infinite sums.
Izzhov said:A closed-form formula is when you take a formula with an infinite sum, such as
[tex]
s = \sum_{k=0}^\infty ar^k
[/tex]
and simplify it to an algebraic formula, which in this case would be
[tex]
s = \frac{a}{1 - r}
[/tex]
(Assuming, in this case, that r < 1.)
Understand?
Kurret said:Couldnt we define a closed formula p(n) over a set of functions (for example the elementary functions, or maybe only the functions f(x)=x, f(x)=c) combined with a certain set of operations (for example ^*/-+) as a formula whose number of terms (functions):
1) is not infinite
2) is not depending on n.
Izzhov said:I go by the Wikipedia definition.
The Partition Function p(n) is a mathematical function that counts the number of ways a positive integer n can be expressed as a sum of positive integers, without regard to order. It is denoted as p(n) or P(n).
The Partition Function p(n) is closely related to number theory, specifically the study of integer partitions. It has applications in areas such as combinatorics, algebra, and modular forms.
Yes, the Partition Function p(n) can be calculated for values of n up to several hundred. However, as n increases, the computation becomes more complex and time-consuming. For very large values of n, approximations and asymptotic methods are used.
Yes, the Partition Function p(n) has practical applications in fields such as physics, chemistry, and computer science. It is used to model and analyze systems with a large number of particles, such as gases and solids, and in the study of prime numbers.
Yes, there are several open research problems related to the Partition Function p(n). These include finding closed-form expressions for p(n), determining the behavior of p(n) as n approaches infinity, and investigating connections between p(n) and other mathematical functions.