DEFINITIONS

Parcelatories, or Partitions, is a mathematical function of Combinatory Analysis which indicates how many possible forms an Whole Number can be obtained from the sum of others smaller Whole Numbers.

An example usually mentioned is the Parcelatories of the number 10.

The number 10 can be obtained from 42 different additions. The list below contains all possible counts:


Unlike the pure combination of elements, the Parcelatories are bounded by the value of the sum of its parts. As can be seen, the five Parcelatories of 10 with 2 parcels (1+9, 2+8, 3+7, 4+6, 5+5) doesn't match the numbers of combination of 10 taken 2 at a time (1+2, 1+3, 1+4, ..., 9+10), precisely because there is the premise that the sum must be equal to 10.

For calculation of Parcelatories we have the following relations:

The total of entire Parcelatories of a number is the sum of all of individual Parcelatories of it. That is,

NUMERICAL OPERATIONS

The studies about Parcelatories are invaluable! They date from the seventeenth century. In 1669, letters from Leibniz to Bernoulli already commented about this.

Lots of features about the Theory of Numbers can be obtained from Parcelatories.

One of the most basic examples is the identification of Perfect Parcelatories, that is, those whose values of the parcels are all equal. In the case of number 10, the Perfect Parcelatories are:

10,
5 +5,
2 +2 +2 +2 +2, 
1 +1 +1 +1 +1 +1 +1 +1 +1

and they correspond exactly to all possible whole multiplications of 10 (10 = 10 * 1, 10 = 2 * 5, 10 = 5 * 2 and 10 = 1 * 10).

Thus, for all Natural Number, Multiplication is nothing more than Perfect Parcelatories!

Similarly, we can define Potentiations by Parcelatories. See below all possible additions of the square of 3 (9) and 4 (16).

   Parcelatories of 9 

   Parcelatories of 16

Although they are numerous, note that the latest parcels of the additions which the number of parcels is equal to the square root of the number (in red) are all exactly equal!This happens with all the squares. So we can say that, for all Natural Numbers, Squares are nothing more than Perfect Parcelatories with an equal number of parcels.

More widely, among Perfect Parcelatories there are a triangular relationship between the number of the parcels (p), the value of the parcels (v) and the value of power (xⁿ). The general equation can be expressed as follows:

As an illustration, we can use the Parcelatories of the square and the cube of 2:

  Parcelatories of 4

  Parcelatories of 8

Note that with the square (2² = 2+2), the number of parcels (p) is equal to two, and the value of the parcels (v) is also equal to two. And p * v = xⁿ, or 2 * 2 = 4.

With the cube (2³ = 4+4 and 2³ = 2+2+2+2), the number of parcels (p) is equal to two and the value of the parcels (v) is equal to four; or the number of parcels (p) is equal to four and the value of the parcels (v) is equal to two. In either case, p * v = xⁿ, or 2 * 4 = 8.

So, we can say that, for all Natural Numbers, Powers are nothing more than Perfect Parcelatories which follow the equation p * v = xⁿ.

With this formula is easy to see that equality between the number of parcels (p) and value of parcels (v) occurs only with squares:

If p = v, then  p * v = p * p = p² = xⁿ. Or, n = 2!

Moreover, if n is greater than 2, always there will be m-ways of found the representation of a Potentiation between the sum of a number raised to a power n. Which m (m > 1) is the number of perfect divisors of the number.

In other words, we can say that within an entire Parcelatory of a number, only we can find additions with the number of parcels equals to the value of the parcels, when the number were a square.

    This is a fundamental condition of Arithmetic with Whole Numbers.

A lot of mathematical problems can be expressed with specific Parcelatories. The difficulty of practical demonstration lies in the fact that it's necessary a very large processing to obtain the Parcelatories of numbers greater than 500.


HERE is an Excel spreadsheet which you can obtain complete Parcelatories since 1 until 25. Although it's possible to calculate the Parcelatories of numbers greater than 25, the process will be incomplete. If you need to expand it or even a class in Java language, you can contact me.