Partition theory is a branch of number theory that studies ways of writing integers as sums of positive integers, disregarding the order of the addends. For example, the partitions of are:

Partition theory is a branch of number theory that studies ways of writing integers as sums of positive integers, disregarding the order of the addends. For example, the partitions of are:

1. 

2. 

3. 

4. 

5. 

Thus, there are 5 partitions of the number .

Ramanujan's Contributions to Partition Theory

Ramanujan introduced deep and innovative results related to partition functions , which represent the number of partitions of . Some of his most famous results include:

1. Asymptotic Formula: He, along with G.H. Hardy, developed an asymptotic formula for , making it easier to approximate the partition function for large numbers.

2. Congruences: Ramanujan discovered remarkable congruences in partition functions:

These congruences show surprising patterns in how partitions behave modulo certain numbers.

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Example 1: Partitions of 

The partitions of are:

1. 

2. 

3. 

4. 

5. 

6. 

7. 

Thus, .

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Example 2: Verifying a Congruence

Let's verify Ramanujan's congruence for .

Step 1: Compute and :

Step 2: Check divisibility:

Thus, Ramanujan's congruence holds true.

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Example 3: Large Partition Approximation

To estimate , we use the Hardy-Ramanujan formula:

p(n) \sim \frac{1}{4n\sqrt{3}} \exp\left(\pi \sqrt{\frac{2n}{3}}\right)

For :

p(100) \sim \frac{1}{400\sqrt{3}} \exp\left(\pi \sqrt{\frac{200}{3}}\right)

This gives an approximation of , which matches the actual value computed through exact methods.

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Significance

Ramanujan's insights into partition theory not only solved mathematical puzzles but also laid the foundation for modern applications in combinatorics, statistical mechanics, and computer science. His results demonstrate the intricate beauty of numbers and their hidden patterns.