PARCELATORIES OF SECOND ORDER

Numbers don't exist in fact. What exists are counting processes, a human ability that consist in to quantify, measure, compare and enumerate objects. Any object!

Because they exist only in our imagination, numbers are only abstract objects we use to calculate our usual counts.

As any abstract object, it was necessary to create symbolic properties to represent them. But substantially, they don't exist! They are therefore, fundamental cognitive concepts.

However the counting is a more fundamental concept yet! So fundamental that we don't perceive it. For this reason, it is viable to define numbers in terms of counts.

The simplest counting is one that happens with the addition of the unit. Any number, except zero, can be expressed as a simple addition of two other numbers. For example, a count of coins can be done as follows:

Zero, that is the beginning;
One coin, which is equal to the unit plus zero (1+0);
Two coins, which is equal to the unit plus one (1+1);
Three coins, which is equal to the unit plus two (1+2);
Four coins, which is equal to the unit plus three (1+3);
Five coins, which is equal to the unit plus four (1+4);
and so on.

Quite simple, this process of counting with two parcels is the most fundamental 'process-concept' of Mathematics. And, precisely, the Parcelatories with 2 parcels — Parcelatories of Second Order or simply 2th Parcelatories — identify and describe any number.

Surely someone will question "If the countings are more fundamental than the numbers, why do we use the numbers and not the countings?" The most sensible explanation is given by the Law of least effort: it is more practical for the human mind to work with numbers than with countings.

The number 5, for example, can be completely defined as the result of two additions with 2 parcels. They are:

2th Parcelatories of 5

Even countings of counts, or in other words, counts with more than two parcels, are also reflexes of counts with two parcels. To explain this, let's take as example the development of the Parcelatory P(13,5):

5th Parcelatories of 13

In its turn,

4th Parcelatories of 12

In the same way:

3th Parcelatories of 11

And so forth.

Note that the first counts, drafted with the red color, correspond to the counts of the parcelatory's first term added to the parcel "1+" in front of they (Examples:1+1+9 and 1+9;1+2+4+5 and 2+4+5;1+1+2+4+5 and 1+2+4+5).

The second counts, drafted with the blue color, correspond to the counts of the parcelatory's second term where all parcels are added to the number 1 (Examples: 2+2+7 and 1+1+6; 2+2+3+3+3 and 1+1+2+2+2; 3+3+3+3 and 2+2+2+2). Because the counts of the second term always keep a equivalent relationship between the parcels and themselves, they are which determine the number's dividers.

So, the number 12 has 3 as a divisor because it appears in all parcels of its Parcelatory P(12,4): 3 + 3 + 3 + 3.

On the other hand, the number 13 has no divisors because in any of its Parcelatories, P(13,1), P(13,2), P(13,3), ... and P(13,12), has no repeated parcels, except the last one (P(13,13)).

Stated another way, what determines whether a number has divisors or not is its parcelatories' second term, that is, if the parcelatory's second term of a number has divisors, so the number has the same dividers as well.

Developing this analysis recursively to smaller numbers, it notes that, if the development of the 2th parcelatory of any number has multiples between its parcels, so the first parcels are divisors of this number.

Finally, with the 2th Parcelatories is possible to identify several mathematical properties of abstract objects called "Numbers", such as:

* Divisors;
* Primality;
* Potentialities.

Set out below are the techniques for determining of these properties.

Divisors of a Number

First it is necessary to explain that 'to extend the 2th parcelatory of a number' consists to create two columns of numbers, the first begins with the unit and ended with half the number, in ascending order; and the second begins with the number minus one and finished at around half the number, in descending order, in such a way that the sum of the parcels results always in the original number.

So, to determine the divisors of a number, simply it must extend its parcelatory of second order and verify when the first is a multiple of the second. If this happens, it is because the first parcel is a multiple thereof.

Follow the example of some interesting parcelatory with two parcels:

Parcelatories of Second Order of the number 10: In this development, it verifies that the divisors of 10 are 1, 2 and 5, because 1 is a divisor of 9, 2 is a divisor of 8 and 5 is divisor of itself. Included to these dividers the number 10 itself what is also divider its own. Thus, 1, 2, 5 and 10 are all the divisors of 10.

Parcelatories of Second Order of the number 13: Ever in this development, it verifies no multiplicity among the parcels, except 1+12. So, it can say that the only divisors of 13 are 1 and 13, or that 13 is a Prime Number.

Parcelatories of Second Order of the number 20: In this case, it verifies that the numbers 1, 2, 4, 5 and 10 are the dividers of 20, besides itself.

This method is much simpler than the traditional method, which is necessary to check if the number is divisible by other smaller numbers, moreover it isn't necessary to divide it by Prime Numbers.

Primality of a Number

The Primality of any number can also be obtained by the method of identification of its divisors, since the Prime Numbers are those which don't have any divisor.

This method is similar to the Primality Test suggested by Indian mathematicians M. Agarwal, N. Kayal and N. Saxena, whose algorithm is now called Primality Test AKS.

Unlike of the determination of the divisors of a number, it isn't necessary to check all the parcelatories of second order to know if a number is prime or not. The process can be greatly facilitated with knowledge of smaller prime numbers.

Potentialities of a Number

The Potentiality of a number can also be obtained by the Parcelatories of Second Order. To do this just extend the 2th parcelatory of a number (n) and verify when any power of the first parcel (p1x) is equal to unit added to the division of second (p2) by it (p1). If this happens, it is because the number (n) is the (x+1)th power the first parcel (p1).

Mathematically we have:
1 + (p2 / p1) = p1x
and
p1 + p2 = n.
i.e.,
p2 = n - p1.
Replacing p2 in the first equation, we have that
1 + ((n - p1) / p1) = p1x
or
(p1 + n - p1) / p1 = p1x,
Which reduces to
n / p1 = p1x.
So,
n = p1(x+1).

Follow the example of the 2th Parcelatories of the number 81:


These, among many other properties, can be obtained directly from the 2th parcelatories of any number. In other words the 2th parcelatories form the skeleton of any number, regardless of its magnitude.

The importance of this affect directly the final result of some Differential Calculus and mainly Integral Calculus, when the parcelatories result in differents kinds of integrations.

HERE is an Excel spreadsheet which can be use to identify the properties of the 'object-numbers' between 1 and 2024. Evidently that the same algorithm can be used to any number, however there are the hardware's limitations. It would be interesting to test these theorems in super-computers. If someone with access to super-computers has conditions and interest in test it, please, let me know.