THE SOLITUDE OF THE PRIME NUMBERS

Prime Numbers can be divided only by 1 and by themselves. They belong to the infinite series of the Integer numbers and they are "compressed" between two or more non prime numbers. They are characterised by a profound beauty since they are difficult to find from a certain point onwards along the series of integers, therefore it may seem that they are "suspicious in essence" and to some degree solitary. Sometimes I ask myself if they are so solitary and more capable of socialising since at a first glance, for some reason, it seems that they are incapable of it. Between the prime numbers there are some even more special: mathematicians call them twins: prime numbers that are at the shortest distance and incapable of touching each other as there is always another number that separates them. Examples of twin primes are found in the 11 and 13, 17 and 19, 41 and 43 to mention some.

The problem that we want to resolve today is to respond the following question: is the family of the prime number solitary or not? and How to judge if the prime numbers are solitary or not?

We know that prime numbers are a family (please note that family is not used to indicate a set, these are 2 different concepts from a mathematical standpoint) of integers and it is difficult to know how many they are.

However if there is the possibility to isolate the "prime numbers family" , how could we decide whether this family is constituted by solitary components or not? This is a great problem of non obvious solution and the Number Theory can help us to find and answer.

The simplest approach would be to infer that if the family is a finite family, in other words composed of a finite number of elements, then those elements or numbers will be solitary in a world with infinite "inhabitants" like that of the Natural Numbers for example. However, given that the prime numbers are infinite (this can be demonstrated and the demonstration is not contemplated in this context), we are interested to determine whether there is the possibility to identify families composed of infinite numbers and determine their degree of solitude..

The definition provided by Math to evaluate the solitude of a family, is very simple and can be conceptualised in the steps below:

1) We take all the elements of the family

2) We invert the elements

3) We sum the inverted elements

4) We evaluate the result

Let's start abstracting this simple concept by considering the set of Natural Numbers and a family of Natural numbers:

Natural Numbers:

1 2 3 4 5 6 ............

A family of Natural numbers constituted by:

a0 a1 a2 a3 ..............

To evaluate the degree of solitude, according to what said above, it is sufficient to apply steps 1), 2), 3), and 4):

A) 1/a0 + 1/a1 + 1/a2 + ....... +

If the sum of the inverted elements of the family is finite and different from the sum of the inverted elements of another family, we can conclude that one family is more solitary than another (Please note that these steps are well defined and applicable if the family is either finite or infinite).

Before articulating further, since the interest of this speculation is to focus on how ideas are articulated cognitively, according to knowledge, and not to the the final "ready-to-apply" result, let's clarify why it is reasonable and convenient to execute the sum of the inverted elements of a family.

For example if we consider the sum of all the Natural numbers we find, thanks to Ramanujan who was accustomed to receive these insights from the Goddess Kali during meditation ( I omit the demonstration), the result below:

B) 1 + 2 + 3 + 4 + 5 + 6 + ....... + = - 1/12

Therefore the reason to sum the inverted of the initial numbers belonging to the family is because, if we are not like Ramanujan, we might let the sum of the non inverted numbers to diverge to infinite and this posits an obstacle to distinguish between infinite families of numbers.

Let's then execute the sum of the inverted Natural numbers and see what happens:

C) 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + ...... +

We need to see if there are extreme situations and the sum above is the most extreme:

If sum C is finite then there is no way to identify the degree of solitude of Integers Number families constituted of infinite numbers. However it is easy to recognise that the sum C represents the harmonic series and its sum is infinite. the demonstration of sum C is non trivial and not matter of this speculation, however it is worth to provide an intuitive approach from which we can infer that this sum is divergent:

Let's consider the first rational term of the sum : 1/2

Looking at the following terms 1/3 and 1/4 we can note that

1/3 + 1/4 > 1/2

Moving forward again we have 1/5 , 1/6, 1/7 and 1/8 and we can note that

1/5 + 1/6 + 1/7 + 1/8 > 1/2

Abstracting the process repetition to the infinite we will always sum longer chunks of numbers (1 number, 2 numbers, 4 numbers, 8 numbers ... 2 to the power of n numbers) which returns a sum greater than 1/2, therefore bringing us to infer that this series is divergent to infinite.

Therefore we can say

`Let's now take another step further by focusing on the family of the power of 2:

2, 4, 8, 16, .......

Let's execute the sum of the inverted numbers:

1/2 + 1/4 + 1/8 + .....+

We know that this sum is finite and equal to 1 (I omit the demonstration as I do not remember it myself and because is laborious)

we have then the result below:

From the above result we can then come to the conclusion the the family of the power of 2 is solitaire or "lonely" against the Integer Numbers.

Let's now try to reflect and elaborate on the question below:

Is the family of the prime numbers like the family of the natural number (not lonely as the sum of the inverted numbers diverges) or like the family of the power of 2 (characterised by some degree of loneliness as the sum of the inverted numbers is finite and equal to one)?

The relevant thing that was demonstrated in the '700 by Eulero, is that if we take the sum of the inverse of the prime numbers, which is our metric to assess the degree of solitude of a family, a sum as such is infinite. The demonstration is not simple. However, there is the possibility to articulate the idea in an intuitive way.

To Demonstrate that the sum of the inverted prime numbers is infinite,

1 + 1/2 + 1/3 + 1/5 + 1/7 + .... + ,

Eulero considered the sum of the inverse of the integers numbers below:

C) 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + ...... +

And made a peculiar connection with the prime numbers to ultimately derive the Theorem below:

THEOREM: There Exist infinite prime numbers

How to Demonstrate this result intuitively?

There is a classic demonstration made by Euclide which shows that if we consider a series of prime numbers it is always possible to find another one that is greater than those belonging to the series. Intuitively, multiplying the prime numbers of a given series of prime numbers and adding 1, we have as result a number that has no common dividers with the prime numbers of the series. As a consequence a number as such will be prime, therefore greater, or will exhibit dividers different from the numbers of the series.

Euler made a different demonstration.

He said: let's negate the predicate and assume, per absurd, that there is only 1 prime number and let's call this number p.

How numbers are composed? As showed by Euclide, all the numbers can be decomposed in prime factors in a unique fashion.

Given that all the numbers can be decomposed in prime factors and we are assuming that there is only one prime number p, the only thing that can change in the numbers is the exponent (power).

So Assuming the existence of a unique prime number, the integer numbers can be represented as below

Let's now assess the degree of solitude of this family of numbers by evaluating the sum of the inverse of the number belonging to the family:

This is a very simple family since each family's number except 0 can be derive by the previous multiplying by 1/p. In other words we have obtained a geometric progression.

We are also aware that all the geometric progressions have a finite sum and the sum equal to

D) p / (p-1)

However result D (a finite sum) is in contraddiction with what demonstrated previously. We have already demonstrated that this sum is infinite. Therefore there cannot be only a unique prime number

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[ HOW to calculate the sum of a geometric progression:

Given the progression

1 + a + a**2 + a**3 + ...... + = S

If a multiply by a this equation we have:

a + a**2 + a**3 + ..... + = aS

By subtracting the 2 equations we obtain the sum as

S = 1 / (1-a) ]

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If there cannot be only one prime number, what happens if we assume , per absurd, the existence of only 2 prime numbers?

Let's assume that p and q, per absurd, are the only 2 existent prime numbers:

Here we have:

E) (1 + 1/p + 1/p**2 + ......) (1 + 1/q + 1/q**2 + .....)

On one side we have the numbers which can be composed of p and on the other those constructed from q.

What is the product of these 2 infinite sums :

In the same way we can come to the conclusion that there cannot be only two prime numbers. Extending the same approach to more prime numbers the same result applies which proves that the prime numbers are infinite. If the sum of the inverse of the integer numbers is infinite it is not possible to have a finite number of prime numbers.

Let's pay attention to the fact that we have simply conceptualised another way of representing the sum below:

1 + 1/2 + 1/3 + ........ +

The above, as already demonstrated is an infinite sum and it is also equal to the infinite product below:

F) 1 + 1/2 + 1/3 + ..... + = (1 + 1/2 + 1/2**2 + 1/2**3 + .....) (1 + 1/3 + 1/3**2 + ...) (1+ 1/5+.....) ....

In other words we transform an infinite sum in an infinite product. This signifies that the operation of sum and product contaminate and confuse with each other when we tend to the infinite.

When we tend to the infinite we assist to the contamination of the opposites and the confusion of the symbols, we are coming back, from a mystical perspective, to the unity or the Monad as Leibnitz would have asserted.

In the end we can re-write the above result F as :

The sum over the integer numbers is equal to the product over the prime numbers

This is the beginning of what is called Analytical Theory of the number .

If we express the power (exponent) in the above equation we can derive the Z function of Riemann

In Conclusion we have found out that the prime numbers are infinite and that their family is not so lonely inside the integer's numbers family.

There is much more to say and I stop it here. Maybe I will continue this speculation in the future by characterising the family of the squared (1, 4, 9, 16, ......)

Eulero demonstrated that the degree of solitude of this family is PI/6