Origin of Sexy Prime Numbers, Origin of Cousin Prime Numbers, Equations from Supposedly Prime Numbers, Origin of the Mersenne Number, Origin of the Fermat Number ()
1. Introduction
The origin of cousin prime numbers and that of sexy prime numbers and their classes are misunderstood by mathematicians and their solution is the subject of mathematical conjecture.
This article has raised a venomous thorn under the feet of scientists, namely the understanding of cousin prime numbers and that of sexy prime numbers.
In this article we have demonstrated the origin and class of sexy primes and those of cousin primes. We have shown that there are two types of sexy prime numbers according to their classes. We have shown in which context a pair of prime numbers which differ from 6 is said to be sexy and in which context it is said to be real sexy.
We have shown that two prime numbers which differ from 4 or even two cousin prime numbers follow one another. An observation made on the supposedly prime numbers allowed us to establish two pairs of equations which are the subject of mathematical conjectures, their resolutions of which are made available to researchers in the field, then we deduced from these equations the origin of the number of Mersenne and the origin of Fermat is number.
The rest of the article is organized as follows, 2. Origin of sexy prime numbers; 3. Origin of cousin prime numbers; 4. Equations from supposedly prime numbers; 5. Origin of the Mersenne number; 6. Origin of the Fermat number; 7. Conclusion followed by bibliography and acknowledgments.
2. Origin of Sexy Prime Numbers
2.1. Definition
Sexy prime numbers are pairs of prime numbers that differ from 6. The real sexy prime numbers are pairs of successive prime numbers that differ from 6 [1] .
Remark:
The interestiong thing about sexy primes is that not only do they form pairs, but also triplets and quardruplets.
As for what concerns us for the remainder of this article, we will focus on pairs of sexy primes.
2.2. Set of Supposed Prime Numbers Noted Esp
According to the Euclidean division theorem for positive integers, we have
,
/
and
We then write:
We can write:
Example:
• If a = 1
• If a = 2
• If a = 3
• If a = 4
• If a = 5
• If a = 6
Consider the set
Note1:
The elements of
are even.
The elements of
are even.
The elements of
are multiples of 3.
The elements of
are even.
When we eliminate these four previous sets in N we are left with the following two sets:
and
.
Consequently the set
contains all the prime numbers except 2 and 3.
Note2:
.
Demonstrate that
Let
with
so
and
Then
.
So:
We note that the set
contains all prime numbers except 2 and 3. Name Esp: The Set of supposedly prime numbers,
contains all prime numbers [2] .
Remark:
Prime numbers other than 2 and 3 are generated by the formulas
and
with
.
A prime number other than 2 and 3 can be written in the form
or
with
.
2.3. Class of a Supposed Prime Number other than 2 and 3
with
then
with
with
then
with
Consequence:
A supposed prime number N is of class U if and only if
with
A supposedly prime number N is of class V if and only if
with
2.4. Arrangement of Supposed Prime Numbers other than 2 and 3
with
and
with
with
with
We have:
, for fixed n.
2.5. Class U Sexy Prime Numbers
with
Consequence:
is prime sexy if and only if
is prime and
is prime
is real prime sexy if and only if
is prime,
is prime and
not prime.
Statement:
Let
with
and
with
sexy primes of class U are the couples
such that
is prime,
is prime.
The Real sexy primes of class U are pairs
such that
is prime,
is prime and
is not prime.
Example:
a) Determine the class of the couple (47; 53)
b) Show that it is sexy
c) Show that it is a real sexy
Solution
a)
with
so 47 is class U
with
so 53 is class U
Hence the couple (47; 53) is class U
b) 47 is prime and 53 is prime (1)
53 − 47 = 6 (2)
(1) and (2) so (47; 53) is prime sexy
c)
then
But 49 is a non-prime number so (47; 53) is a real sexy (47 and 53 follow one another) and class U.
Counter-example:
a) Determine the class of the couple (5; 11)
b) Show that it is sexy
c) Show that it is not a real sexy
Solution:
a)
with
so 5 is class U
with
so 11 is class U
Hence the couple (5; 11) is of class U.
b) 5 is prime and 11 is prime (1)
11 − 5 = 6 (2)
(1) and (2) so (5; 11) is prime sexy
c)
then
But 7 is a prime number so (5; 11) is not a real sexy prime because 5 and 11 do not follow each other (there is 7 in the middle).
2.6. Class V Sexy Prime Numbers
with
Consequence:
is prime sexy if and only if
is prime and
is prime.
is real prime sexy if and only if
is prime and
is prime and
not prime.
Statement:
Let
with
and
with
sexy primes of class V are the couples
such that
is prime,
is prime.
The Real sexy primes of class V are pairs
such that
is prime,
is prime and
is not prime.
Noticed:
, for fixed n.
Example:
1. Determine the class of the couple (61; 67)
2. Show that it is sexy
3. Show that it is a real sexy
Solution:
a)
with
so 61 is class V
with
so 67 is class V
Hence the couple (61; 67) is class V
b) 61 is prime and 67 is prime (1)
67 - 61 = 6 (2)
(1) and (2) so (61; 67) is prime sexy
c)
then
But 65 is a non-prime number so (61; 67) is a real sexy prime of class V.
Counter-example:
a) Determine the class of the couple (7; 13)
b) Show that it is sexy
c) Show that it is not a real sexy
Solution:
a)
with
so 7 is class V
with
so 13 is class V
Hence the couple (7; 13) is class V
b) 7 is prime and 13 is prime (1)
13 − 7 = 6 (2)
(1) and (2) so (7; 13) is prime sexy of class V
c)
then
But 11 is a prime number so (7; 13) is not real sexy prime because 7 and 13 do not follow each other (there is 11 in the middle).
3. Origin of Prime Cousin Numbers
3.1. Definition
Cousin prime numbers are pairs of prime numbers that differ from 4 [3] .
3.2. Cousin prime numbers
with
and
with
with
Consequence:
is cousin prime if and only if
is prime and
is prime.
Statement:
Let
with
and
with
cousin primes are pairs
such that
is prime,
is prime.
Noticed :
, for n fixed therefore
and
are two supposedly prime which therefore follow each other:
If
is cousin prime then
and
are two prime numbers that follow each other.
Example:
The following three pairs are cousin prime pairs (7; 11), (13; 17), (19; 23).
Counter-example:
When two prime numbers do not follow each other they cannot be cousins. To give counter examples, simply choose two non-successive prime numbers.
4. Equations from Supposedly Prime Numbers
4.1. Second form of Writing the Set of Supposed Prime Numbers
Consider
the set of supposed prime numbers.
Supposedly prime numbers other than 2 and 3 are generated by the following two formulas:
with
and
with
Noticed:
• Let
with
therefore
and
so:
• Let
with
therefore
and
therefore
.
So
From where
or
Noticed:
with
and
with
We have:
and
with
.
Consequence :
A number N is a supposed prime number other than 2 and 3 if and
Only if
or
4.2. Observation
We found through calculations that:
1. The formula
with
only gives supposedly prime numbers.
2. The formula
with
, with the exception of 1, only gives supposedly prime numbers.
3. Formulas
with
do not all give supposed prime numbers.
NB:
This observation was made through in-depth calculations up to the supposedly very high first ones but nothing proves it to us through demonstrations.
Let us accept that this hypothesis is true, we will therefore have
is a subset of the set of presumed prime numbers denoted by
.
Result :
1.
with
are supposed primes other than 2 and 3 so
with
is written in the form
with
or
with
.
2.
with
are supposed prime numbers other than 2 and 3 more
gives 1 for n = 1, m = 1 or n = 3, m = 2 and 6n + 1 gives 1 for n = 0 therefore
with
can be written as
with
or
with
.
States :
• For all non-zero integers n and m, there exists a non-zero integer k verifying the equation:
or
• For all non-zero integers n and m, there exists a non-zero integer k and an integer p verifying the equation:
or
These two pairs of equations are made available to researchers in the field for their demonstrations.
Noticed:
• The formula
with
gives an infinity of prime numbers and an infinity of non-prime numbers.
• The formula
with
, with the exception of 1, gives an infinity of prime numbers and an infinity of non-prime numbers.
• Not all prime numbers are written in the form
with
.
NB: (for verification)
1. To verify that the formula
with
, gives only supposed primes, just choose two integers
and
then:
• Calculate
which gives
• Check that:
or
Example:
if n = 2; m = 2
so 5 is a supposedly prime
if n = 2; m = 3
so 23 is a supposedly prime
if n = 2; m = 4
so 77 is a supposedly prime.
We can continue the verifications by assigning larger numbers to n and m, and we will see that we will always have supposedly prime numbers.
2. To verify that the formula
with
, gives only supposed primes, just choose two integers
and
then:
• Calculate
which gives
• Check that:
or
Example:
if n = 1, m = 1
and 5 is a supposedly prime number
if n = 2, m = 2
so 13 is a supposedly prime number
if n = 2, m = 3
so 31 is a supposedly prime number
if n = 2,m = 4
so 85 is a supposedly prime number.
We can continue the verifications by assigning larger numbers to n and m, and we will see that we will always have supposedly prime numbers.
5. Origine of the Mersenne Number
5.1. Definition
In mathematics and more precisely in arithmetic, a Mersenne number is a number of the form
(often denoted
), where n is a natural number not zero; a Mersenne prime (or prime Mersenne number) is therefore a prime number of this form. These numbers owe their name to the erudite French religious and mathematician Marin Mersenne of the 17th century; but, nearly 2,000 years earlier, Euclid was already using them to study perfect numbers. Before Mersenne, and even for some time after him, the search for Mersenne primes is intrinsically linked to that of perfect numbers [4] .
5.2. Demonstration of the Origin of the Mersenne Number
Consider the formula
with
.
Let
with
.
So this formula generates an infinity of supposed prime nombers and an infinity of multiple numbers of 2 or/and 3.
If we set m = 0, we will have
.
for
therefore
with
, this is the Mersenne number.
6. Origin of the Fermat Number
6.1. Definition
A Fermat number is a number that can be written in the form
, with n a natural number. The Fermat number of rank n,
, is denoted
.
The sequence (
), which begins with 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617 is listed as sequence A000215 in the EIS.
These numbers owe their name to Pierre de Fermat, who made the conjecture that all of these numbers were prime. This conjecture turned out to be false, F5 being composite, as are all the following ones up to
. It is not known whether the numbers from F33 onwards are prime or composite. Thus, the only known Fermat primes are five in number, namely the first five
and
, which are respectively 3, 5, 17, 257 and 65,537.
Fermat numbers have interesting properties, generally arising from modular arithmetic. In particular, the Gauss-Wantzel theorem establishes a link between these numbers and the construction with a ruler and compass of regular polygons: a regular polygon with n sides can be constructed with a ruler and compass if and only if n is a power of 2, or the product of a power of 2 and distinct Fermat primes [5] .
6.2. Demonstration of the Origin of Fermat’s Number
Consider the formula
with
.
Let
with
.
So this formula generates an infinity of supposed prime nombers and an infinity of multiple numbers of 2 or/and 3.
If we set m = 0 and we choose n = 2k with
.
We will have:
with
, this is Fermat’s number.
Noticed:
Concerning the origins of other formulas of the past, it will be necessary to make a limited development of
with
.
7. Conclusion
The formulas established in this article have penetrated the secret causes of prime cousin numbers, sexy prime numbers, the Mersenne number and those of the Fermat number. The two pairs of equations established in the article are objects of mathematical conjectures whose resolutions are available to mathematicians. I plan to publish another article very soon which will focus on the pseudo-periodicity of prime numbers.
Biography
Mady NDIAYE, mathematics professor at CEM Badara CEM Bdara Mbaye Kaba, IEF Grand Dakar, IA Dakar, Ministry of National Education of Senegal. Dakar, Senegal.
Born on May 21, 1980 in Sinthiou maleme (Senegal).
Acknowledgement
I thank:
Prof. Mamadou Barry, Department of Mathematics and Computer Science, Faculty of Science and Technology, Cheikh Anta Diop University of Dakar (UCAD).
Prof. Mamadou Sangar, Department of Mathematics and Computer Science, Faculty of Science and Technology, Cheikh Anta Diop University of Dakar (UCAD).
Prof. Aboubaker Shedikh Beye, Department of Physics, Faculty of Science and Technology, Cheikh Anta Diop University of Dakar (UCAD).
Prof. Omar Sakho, Department of Physics, Faculty of Science and Technology, Cheikh Anta Diop University of Dakar (UCAD).
Dr. Abdoulaye Diop, Department of Physics, Faculty of Science and Technology, Cheikh Anta Diop University of Dakar (UCAD).
All the doctors and doctoral students who attended my presentations by observations and suggestions.