Utente:Grasso Luigi/sandbox4/Funzione di partizione
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In number theory, the partition function Template:Math represents the number of possible partitions of a non-negative integer Template:Mvar. For instance, Template:Math because the integer 4 has the five partitions Template:Math, Template:Math, Template:Math, Template:Math, and Template:Math.
No closed-form expression for the partition function is known, but it has both asymptotic expansions that accurately approximate it and recurrence relations by which it can be calculated exactly. It grows as an exponential function of the square root of its argument. The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument.
Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences. For instance, whenever the decimal representation of Template:Mvar ends in the digit 4 or 9, the number of partitions of Template:Mvar will be divisible by 5.