What is Srinivasa Ramanujan partition theory?

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 n ends in the digit 4 or 9, the number of partitions of n will be divisible by 5.

What is the partition formula?

A partition of a number is any combination of integers that adds up to that number. For example, 4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1, so the partition number of 4 is 5. It sounds simple, yet the partition number of 10 is 42, while 100 has more than 190 million partitions.

What is the partition of 200?

Partition

What is the partition of 5?

Using the usual convention that an empty sum is 0, we say that p0=1. Example 3.3. 2 The partitions of 5 are 54+13+23+1+12+2+12+1+1+11+1+1+1+1.

Who invented partition maths?

Ramanujan noticed that whole numbers can be broken into sums of smaller numbers, called partitions. The number 4, for example, contains five partitions: 4, 3+1, 2+2, 1+1+2, and 1+1+1+1.

How many partitions of 3 are there?

Thus the partitions of 3 are 1+1+1, 1+2 (which is the same as 2+1) and 3. The number of partitions of k is denoted by p(k); in computing the partitions of 3 we showed that p(3)=3.

What is the partition of 10?

There are forty-two partitions of 10. The numbers of partitions of 10 with largest part {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} are respectively {1, 5, 8, 9, 7, 5, 3, 2, 1, 1}. (There are 20 partitions of 10 with largest part odd and 22 partitions of 10 with largest part even.)

How many partitions of 7 are there?

15 such
List all the partitions of 7. Solution: There are 15 such partitions. 7, 6+1, 5+2, 5+1+1, 4+3, 4+2+1, 4+1+1+1, 3+3+1, 3+2+2, 3+2+1+1, 3+1+1+1+1, 2+2+2+1, 2+2+1+1+1, 2+1+1+1+1+1, 1+1+1+1+1+1+1.

What is the partition of 6?

The eleven partitions of 6 are: 6, 5+1, 4+2, 4+1+1, 3+3, 3+2+1, 3+1+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1, and 1+1+1+1+1+1. (b). Since 288 = 32 9 = 25 32 there are 7 2 = 14 such groups. For example, Z32 Z9, Z8 Z4 Z3 Z3 , and Z4 Z4 Z2 Z3 Z3 .

What are mathematical partitions used for?

Partitioning is used to make solving maths problems involving large numbers easier by separating them into smaller units. For example, 782 can be partitioned into: 700 + 80 + 2. It helps kids see the true value of each digit.

How many partitions does 4 have?

The number of partitions of n is given by the partition function p(n). So p(4) = 5. The notation λ ⊢ n means that λ is a partition of n.

What is the partition of 7?

1. List all the partitions of 7. Solution: There are 15 such partitions. 7, 6+1, 5+2, 5+1+1, 4+3, 4+2+1, 4+1+1+1, 3+3+1, 3+2+2, 3+2+1+1, 3+1+1+1+1, 2+2+2+1, 2+2+1+1+1, 2+1+1+1+1+1, 1+1+1+1+1+1+1.

What are the partitions of 8?

Among the 22 partitions of the number 8, there are 6 that contain only odd parts:

• 7 + 1.
• 5 + 3.
• 5 + 1 + 1 + 1.
• 3 + 3 + 1 + 1.
• 3 + 1 + 1 + 1 + 1 + 1.
• 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1.

What are the partitions of 10?