We consider prime numbers, rightly, as
elementary particles of all numbers. Meanwhile, the situation is that we know
(almost) everything about numbers, very precisely, but we have only “empirical”
knowledge and various hypotheses about prime numbers themselves that can only
be obtained by calculations. It is as if something is blocking the paths of
omnipotent theoretical research to the mystical primes. Although this is
another topic (which you will want to return to.)

We would now consider four
related hypotheses giving to this some “empirical” statements.

A well-known concept is the
twin prime. Here, care must be taken that the term itself can be applied to
both a number and a pair of numbers, and this requires attention. So, an n
number is a twin prime if prime and either n-2 or n+2 is prime also. That is,
there is another prime “near” n. Two numbers n and m are twin primes or twin
prime pairs if both are prime and n-m=2 or m-n=2. The topic and wordings can be
complicated by the strange beginning of the primes (2, 3, 5, 7). This “weird
start” is also a separate topic (which is also worth returning to). Now we can
do this by counting the twin prime numbers from 5 onwards. And indeed, the
first “regular” twin pairs: (5,7), (11,13), (17,19), and so on.

Well, these twin primes are of
huge interest, many deal with them, you can read a lot about them. But what are
non-twin prime numbers like? Many times, if there is a highlighted subset,
there may be several others besides it, but this is not the case here (but like
the concept of an even number: if a number is not even, it is odd, and that’s
it). Indeed, if there is no other prime “near” a prime number, then there is
not, and that is it. It seems legitimate to call such primes alone or lonely.
That is, not every prime or twin, or lonely, third case. The first reflexive
question in arithmetic is that there are an infinite number of these? Well,
let's not wait for a theoretical, demonstrable result. However, we can have a
hypothesis.

**H1
(Small twin prime hypothesis)**
There are infinitely many twin primes.

**H2
(Small one-prime hypothesis)**
There are infinitely many one-primes.

The first hypothesis has been
known for a long time, it must have been formulated when we first talked about
twin primes. All that can be added is that many have tried to prove it, and
often partial results are reported (a partial result of the siege of an old
girl may also be that she is already half-virgin, but here we are still far
from it). The question, however, is really interesting because the statement
can by no means be said to be obvious.

In contrast, questioning
Hypothesis H2 would be pure absurdity. Anyone can rightly say: this statement
is infinitely trivial. Indeed. At a glance. But let me prove it! I find the
proof of this statement as hopeless as that of H1, and that is why it deserves
the status of a written hypothesis. In the meantime, I do not rule out that
this was first described in this form.

Now let’s move on to examining
these two animal species.

An obvious and legitimate
question is: if these two kinds of prime numbers va, and presumably there are
infinitely many of both, how are they distributed? It immediately becomes
apparent that the twins are starting strongly: the six primes starting with 5
are all twins, but they are also in good numbers when viewed up to 100. First,
however, let us note that, except for the departure curiosity, where 3, 5, and
7 are no three consecutive twins, triple twins, which is - easily - provable.

**Lemma
1.** If p > 3 and p and p + 2 are prime, p + 4 is not prime.

As a result, you need to
“rest” after each pair of twin primes. All right, come the rest, but the rest
alone does not decide whether it is followed by a twin or a single. For
example, after 7 there is a twin, after 19 there is single. There is nothing
surprising in this that twin prime pairs can form groups. It is not too
difficult to map these. This way we can create sequences that identify (with
their first member) the different groups. A more difficult question is how
large these groups can be. It is quite understandable that the larger such
groups we search for, the less often we find such. And here the difficulties
arise surprisingly quickly. Even the discovery of groups of 5 pairs of twin
primes requires effective computer assistance. Currently, the research
competition is stuck in groups of 10. Sure, we cannot doubt that our
ever-growing computers are making further progress, discovering groups 11 and
12, but the members of these are already inconceivable numbers.

Yet, the human mind does not
reach that much. The question is: for any n numbers, is there a group
containing n pairs of twin primes? Let's not expect a theoretical result for
this either, but let us also have the courage to formulate the third hypothesis:

**H3
(Big Twin Hypothesis)**
For any natural n number, there exists an n-member group of prime pairs.

Before we go any further,
let’s take a look at some of the series related here

Starting numbers of twin prime
pairs:

3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137,
149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599,
617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151,
1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1607 (A001359)

Starting numbers for twin-pair
groups:

5, 11, 101, 137, 179, 191, 419, 809, 821, 1019,
1049, 1481, 1871, 1931, 2081, 2111, 2969, 3251, 3359, 3371, 3461, 4217, 4229,
4259, 5009, 5651, 5867, 6689, 6761, 6779, 6947, 7331, 7547 (A053778)

Summary - Starting numbers for
n-member twin-pair groups:

3, 5, 5, 9419, 909287, 325267931, 678771479,
1107819732821, 170669145704411, 3324648277099157 (A111950, In 2011, Gábor Lévai
announced that he had found a group of 11)

Now let’s turn our attention
to the somewhat marginalized single primes. First, let's say the fourth
promised hypothesis:

**H4
(Big one-prime hypothesis)**
For any n natural numbers there exists a n-member one-prime group.

Undoubtedly, this is an
“obvious” statement like H2, but here it can be said that it does not seem to
be formally provable. Nevertheless, here we can make calculations similar to
twin primes, which are somewhat less common in the literature for single
primes.

The single primes:

23, 37, 47, 53, 67, 79, 83, 89, 97, 113, 127,
131, 157, 163, 167, 173, 211, 223, 233, 251, 257, 263, 277, 293, 307, 317, 331,
337, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 439, 443, 449, 457, 467,
479, 487, 491, 499, 503, 509, 541, 547, 557, 563 (A007510, although this - in
my humble opinion - incorrectly includes 2 as well)

Starting number of double
single prime groups:

47, 79, 83, 89, 113, 127, 157, 163, 167, 211,
251, 257, 293, 317, 331, 353, 359, 367, 373, 379, 383, 389, 397, 401, 439, 443,
449, 467, 479, 487, 491, 499, 503, 541, 547, 557, 577, 587, 607, 647 (A126095)

And for the summary of
single-prime groups, i.e., the starting number of n-membered single-prime
groups:

23, 47, 79, 79, 353, 353, 353, 353, 353, 673,
673, 673, 673, 673, 673, 673, 673, 8641

Before concluding on this
topic, I would like to point out a useful fact. To do this, define two
concepts: if p is a prime number, we call foreground the difference between p
and the previous prime, the background is the differences between the following
prime and p, denoting them FG(p), resp. BG(p). Example: , FG(13) = 2 and BG(13)
= 4

Obviously:

**Lemma
2.** For any prime p
greater than 3, FG(p) and BG(p) are even numbers.

The following is less trivial,
but very useful:

**Lemma
3.** For any prime p,
if , FG(p) = BG(p), then this number is 6 or some multiple of it.

The distribution of the two
types of primes is best shown in the following series, where the numbers are
the “isolation index” of the primes, which is the minimum of FG(p) -1 and BG(p)
-1, i.e. the smaller of the number of composites before and the number of the
composites after p. It follows that (with the exception of 3) a prime is a twin
prime if and only if it has an isolation index of 1. Otherwise, it is a
solitary prime, which is more solitary the higher its isolation index.

0, 0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 3, 5,
1, 1, 3, 1, 1, 3, 3, 5, 3, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 5, 3, 3, 5, 1, 1,
1, 1, 1, 1, 11, 3, 1, 1, 3, 1, 1, 5, 5, 5, 1, 1, 3, 1, 1, 9, 3, 1, 1, 3, 5, 5,
1, 1, 3, 5, 5, 5, 3, 3, 5, 3, 3, 7, 1, 1, 1, 1, 3, 3, 5, 3, 1, 1, 3, 7, 3, 3, 3,
3, 5, 1, 1, 5, 5, 5, 5, 1, 1, 5, 5, 5, 1, 1, 5, 3, 1, 1, 9, 1, 1, 3, 5, 1, 1,
3, 3, 5, 7, 7, 7, 7, 5, 5, 3, 3, 5, 3, 3, 3, 3, 9, 9, 1, 1, 1, 1, 1, 1, 9, 3,
1, 1, 3, 3, 1, 1, 3, 3, 3, 7, 7, 3, 3, 5, 5, 3, 3, 5, 5, 5, 5, 3, 3, 1, 1, 1,
1, 5, 1, 1, 1, 1, 5, 3, 1, 1, 3, 5, 5, 5, 5, 5, 1, 1, 7, 7, 5, 5, 5, 7, 3, 3,
5,

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