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Partition function (number theory)
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<p>In <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, the partition function <i>p</i>(<i>n</i>) represents the <a href="/facts/Number/auQSAE2m">number</a> of possible <a href="/facts/Integer_partition/LtyXWqwH">partitions</a> of a non-negative integer n. For instance, <i>p</i>(4) = 5 because the integer 4 has the five partitions 1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 3, 2 + 2, and 4. </p><p>No <a href="/facts/Closed-form_expression/y0xCXlSk">closed-form expression</a> for the partition function is known, but it has both <a href="/facts/Asymptotic_analysis/kt87qLpu">asymptotic expansions</a> that accurately approximate it and <a href="/facts/Recurrence_relation/JolXC71M">recurrence relations</a> by which it can be calculated exactly. It grows as an <a href="/facts/Exponential_function/ewlYxvR2">exponential function</a> of the <a href="/facts/Square_root/AfrzfBdQ">square root</a> of its argument. The <a href="/facts/Multiplicative_inverse/yczdYyCw">multiplicative inverse</a> of its <a href="/facts/Generating_function/K7UVjnjL">generating function</a> is the <a href="/facts/Euler_function/y6lE3VIL">Euler function</a>; by Euler's <a href="/facts/Pentagonal_number_theorem/z8TSDYAY">pentagonal number theorem</a> this function is an alternating sum of <a href="/facts/Pentagonal_number/P8pNBERv">pentagonal number</a> powers of its argument. </p><p><a href="/facts/Srinivasa_Ramanujan/HBPr4bzd">Srinivasa Ramanujan</a> first discovered that the partition function has nontrivial patterns in <a href="/facts/Modular_arithmetic/0wtRa5dU">modular arithmetic</a>, now known as <a href="/facts/Ramanujan%2527s_congruences/HhcD7AFu">Ramanujan's congruences</a>. For instance, whenever the decimal representation of n ends in the digit 4 or 9, the number of partitions of n will be divisible by 5. </p>