Ramanujan, an expert in number theory. We will see, one of his works.
3= 3
3 = 2+1
3 = 1+1+1
Here 3 is written in three ways. A natural number is expressed as sum of natural numbers.
4= 4
4 = 3+1
4 = 2 + 2
4 = 2+1+1
4 = 1+1+1+1
Here 4 is expressed in 5 different ways.
We define "partition number" p(n) as the number of ways a given natural number 'n' can be written as a sum of natural numbers.
Hence p(3)=3
p(4)=5
It seems easy to find out the partition number. But as the number gets larger, everything goes out of hand. Partition number for 1 to 10 is given below.
n p(n) n p(n)
1 1 6 11
2 2 7 15
3 3 8 22
4 5 9 30
5 7 10 42
p(n) goes on increasing exponentially. Is there a procedure or a formula to find the partition number of 'n'? Ramanujan gave 'almost a correct formula' to calculate p(n).
That is
p(n) = 1/[4 pi root of 3] *eˆ[pi root of 2n/3]
The accuracy of formula increases as the n increases.
for example
p(100) = 190569292,
It is exactly correct.
Partion number formula eluded mathematicians for centuries. Ramanujan made a breakthrough. This formula (paper) is mainly responsible for his election as a follow of the Royals society.
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