Partitions...straightforward

Number Theory Level 2

A partition can be defined as a way of writing a positive integer as a sum of positive integers, where the order of the numbers in the sum doesn't matter.

For example; the number of partitions of 3 is 3 namely:

$\implies 3$

$\implies 2+1$

$\implies 1+1+1$

Similarly, the number of partitions increases with the number. Here are the partitions of the first 6 numbers:

$n \implies \text{number of partitions} \\ 1 \implies 1 \\ 2 \implies 2 \\ 3 \implies 3 \\ 4 \implies 5 \\ 5 \implies 7 \\ 6 \implies 11$

What is the approximate number of partitions of 100 ?

\approx

10,000,000

\approx

100,000,000

\approx

2,000,000

\approx

1,000,000

\approx

200,000,000

2 solutions

Syed Hamza Khalid
Nov 24, 2018

This is a straight application of the Hardy-Ramanujan formula :

Let the number of partitions be $p(n)$ , then : $p(n) \approx \dfrac{1}{4n \sqrt{3}} \exp \Bigg ( \pi \sqrt{\dfrac{2n}{3}} \Bigg )$

Inputting $\boxed{ n = 100 }$ will result in $190,280,893 \approx 200,000,000$

Note: $\exp(x) = e^x$

X X
Nov 23, 2018

Let the number of partitions of $n$ be $p(n)$ , then the approximation of $p(n)$ is $\dfrac{e^{(\pi\sqrt{\frac{2n}3})}}{4n\sqrt{3}}$ , put in $n=100$ will get the answer.

Or you can see OEIS, it is equal to 190569292.

What is OEIS?

Syed Hamza Khalid - 2 years, 6 months ago

On-line Encyclopedia of Integer Sequences

X X - 2 years, 6 months ago

Partitions...straightforward

2 solutions

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