Goldbach Conjectur and Composite Numbers

 

Dr Jagjit Singh

Department of Mathematics, Govt. College Chowari  District Chamba (H.P.)-176302, India

*Corresponding Author E-mail: jagjitsinghpatial@gmail.com.

 

 

ABSTRACT:

In this paper, the results that every even composite number can be expressed as sum of two odd primes, every odd composite number can be expressed as a sum of an odd prime and a number of the form 2n where n  and every odd prime number greater than 3 may be expressed as a sum of a pure number and a number of the form 2n where n  are being presented. Further, for any sufficiently large natural number n, it is presented that every even number up to n can be expressed as a sum of two primes less than n, every even number up to n can be expressed as a sum of two primes less than  and every even number up to n can be expressed as a sum of two primes less than.

 

INTRODUCTION:

Goldbach [1] conjectured that every even number greater than equal to 4 can be expressed as sum of two primes and which has been verified for sufficiently large even numbers but still couldn’t be proved.

 

Schnirelmann [4] gave the result that there is a number k such that every number n is the sum of k or fewer primes:

     (for sufficient large value of n).

Vinogradov [5] proved that from some point on every odd number is the sum of three primes:

      (for sufficient large value of n).

Renyi [3] proved that there is a number k such that for some point on every even number can be written as a sum of prime plus another number which has no more than k prime factors:

                           (Where a has no more than k prime factors and n is sufficiently large even number)

 

3. PURE NUMBERS AND STRENGTH OF EVENNESS:

A number which can be put in the form, where p is a prime, is called pure number.

Every prime number is a pure number but every pure number is not a prime. A pure number is a prime if n=1, and a pure number is composite if n≥2.

Strength of evenness of an even number n may be defined as the natural number  if  where m is an odd number.

PRINCIPLE OF IRREGULARTY OF PRIME NUMBERS:

Prime numbers are always distributed irregularly under any sequence of natural numbers.

Therefore, principle of irregularity of prime numbers may be defined as: there exists at least one prime number in the sequence of natural number defined by  where  is a prime number,  2n≥6 and .

There exist at least one prime number and one composite number in the sequence of natural number defined by  where  is a prime number,  2n≥12 and .

THEOREM-5:

Every even number 2n≥6 can be expressed as sum of two odd primes.

Proof: By the principle of irregularity of primes, there exists at least one prime number in the sequence of natural number defined by  where  is a prime number,  2n≥6 and . Let this one prime number is  therefore   or  i.e. every even number 2n≥6 can be expressed as sum of two primes.

Some even numbers are operated according to theorem-5 in the following analysis;

   Therefore number of primes is 1 when prime number 3 is subtracted from 6 and no composite number is obtained when prime numbers from 3 to  are subtracted from 6.

  8–3=5, 8–5=3; therefore number of primes obtained is 2 and number of composites is nil when prime numbers from 3 to 8–3 are subtracted from 8.

  10–3=7, 10–5=5, 10–7=3; therefore number of primes obtained is 3 and number of composites is nil when prime numbers from 3 to 10–3 are subtracted from 10.

  12–3=9, 12–5=7, 12–7=5; therefore number of primes obtained is 2 and number of composites is 1 when prime numbers from 3 to 12–3 are subtracted from 12.

  14–3=11, 14–5=9, 14–7=7, 14–11=3; therefore number of primes obtained is 3 and number of composites is 1 when prime numbers from 3 to 14–3 are subtracted from 14.

  16–3=13, 16–5=11, 16–7=9, 16–11=5, 16–13=3; therefore number of primes obtained is 4 and number of composites is 1 when prime numbers from 3 to 16–3 are subtracted from 16.

  18–3=15, 18–5=13, 18–7=11, 18–11=7, 18–13=5; therefore number of primes obtained is 4 and number of composites is 1 when prime numbers from 3 to 18–3 are subtracted from 18.

  20–3=17, 20–5=15, 20–7=13, 20–11=9, 20–13=7, 20–17=3; therefore number of primes obtained is 4 and number of composites is 2 when prime numbers from 3 to 20–3 are subtracted from 20.

  22–3=19, 22–5=17, 22–7=15, 22–11=11, 22–13=9, 22–17=5, 22–19=3; therefore number of primes obtained is 5 and number of composites is 2 when prime numbers from 3 to 22–3 are subtracted from 22.

  24–3=21, 24–5=19, 24–7=17, 24–11=13, 24–13=11, 24–17=7, 24–19=5; therefore number of primes obtained is 6 and number of composites is 1 when prime numbers from 3 to 24–3 are subtracted from 24.

  26–3=23, 26–5=21, 26–7=19, 26–11=15, 26–13=13, 26–17=9, 26–19=7, 26–23=3; therefore number of primes obtained is 5 and number of composites is 3 when prime numbers from 3 to 26–3 are subtracted from 26.

  28–3=25, 28–5=23, 28–7=21, 28–11=17, 28–13=15, 28–17=11, 28–19=9, 28–23=5; therefore number of primes obtained is 4 and number of composites is 4 when prime numbers from 3 to 28–3 are subtracted from 28.

  30–3=27, 30–5=25, 30–7=23, 30–11=19, 30–13=17, 30–17=13, 30–19=11, 30–23=7; therefore number of primes obtained is 6 and number of composites is 2 when prime numbers from 3 to 30–3 are subtracted from 30.

  32–3=29, 32–5=27, 32–7=25, 32–11=21, 32–13=19, 32–17=15, 32–19=13, 32–23=9, 32–29=3; therefore number of primes obtained is 4 and number of composites is 5 when prime numbers from 3 to 32–3 are subtracted from 32.

  34–3=31, 34–5=29, 34–7=27, 34–11=23, 34–13=21, 34–17=17, 34–19=15, 34–23=11, 34–29=5, 34–31=3; therefore number of primes obtained is 7 and number of composites is 3 when prime numbers from 3 to 34–3 are subtracted from 34.

  36–3=33, 36–5=31, 36–7=29, 36–11=25, 36–13=23, 36–17=19, 36–19=17, 36–23=13, 36–29=7, 36–31=5; therefore number of primes obtained is 8 and number of composites is 2 when prime numbers from 3 to 36–3 are subtracted from 36.

  38–3=35, 38–5=33, 38–7=31, 38–11=27, 38–13=25, 38–17=21, 38–19=19, 38–23=15, 38–29=9, 38–31=7; therefore number of primes obtained is 3 and number of composites is 7 when prime numbers from 3 to 38–3 are subtracted from 38.

  40–3=37, 40–5=35, 40–7=33, 40–11=29, 40–13=27, 40–17=23, 40–19=21, 40–23=17, 40–29=11, 40–31=9, 40–37=3; therefore number of primes obtained is 6 and number of composites is 5 when prime numbers from 3 to 40–3 are subtracted from 40.

All this can be mentioned by the following Table-1:

Let  where  is a prime number, 2n≥6, and .

Table-1

Even number=2n

Number of prime

Number of composite

Even number=2n

Number of prime

Number of composite

Even number=2n

Number of prime

Number of composite

6

1

0

38

3

7

70

10

8

8

2

0

40

6

5

72

12

6

10

3

0

42

8

3

74

9

10

12

2

1

44

6

6

76

10

10

14

3

1

46

7

6

78

14

6

16

4

1

48

10

3

80

8

12

18

4

1

50

8

6

82

9

12

20

4

2

52

6

8

84

16

5

22

5

2

54

10

4

86

9

13

24

6

1

56

6

9

88

8

14

26

5

3

58

7

8

90

18

4

28

4

4

60

12

3

92

8

15

30

6

2

62

5

11

94

9

14

32

4

5

64

10

7

96

14

9

34

7

3

66

12

5

98

6

17

36

8

2

68

4

13

100

12

12


DISTANCE BETWEEN TWO CONSECUTIVE PRIMES:

We know that difference between two consecutive primes like 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41…etc.  is 2, 4, 6, 8. Further analysis about the distances between two consecutive prime numbers is given by this result.

 

THEOREM-6

The difference between any two consecutive prime numbers may be any even number tending to infinity.

Proof:  Let  be the product of first n primes. Then may or may not be a prime number and none of the numbers  is prime and  may or may not be prime. If both  and  are primes then the distance between two consecutive primes is equal to  otherwise in case if one of the   and   or both of them are not prime numbers then the distance between two consecutive prime numbers may be ≥ But may be infinitely large and therefore difference between any two consecutive prime numbers may be any even number and it tends to infinity when tends to infinity. 

 

If we look at the tables of primes carefully we find that the difference between two consecutive prime numbers is 2, 4, 6, 8,10,… it can be easily verified like this 5–3=2, 11–7=4, 29 –23=6, 97–89=8, 149 –139=10, 211–199=12, 127–113=14,1847–1831=16, 541–523=18, 907–887=20, 1973–1951=22, 1693–1669=24, 2503–2477=26, 2999–2971=28, 4327–4297=30 5623–5591=32, 1361–1327=34 and so on therefore the difference between two consecutive primes will be each even number 2, 4, 6, … Further, we find that for each even number, there exist infinite pairs of prime  numbers such that the difference between two numbers of each pair is that even number. For example, 2=5–3=7–5=13–11=19–17=…, 4=11–7=17–13=23–19=41–37=…, 6=29–23=37–31=53–47=59–53=…, 8=97–89=367–359=387–389=409–401=…, 10=149–139=191–181=251–241, 293–283=…, 12=211–199=223–211=479–467=521–509=…, 14=127–113=307–293=331–317=787–773=…, 16=1847–1831=1949–1933=2129–2113=2237–2221=…, 18=1087–1069=1277–1259=1399–1381=1777–1759=…, 20=907–887=1657–1637=3109–3089=3433–3413=5737–5717=…,22=1151–1129=1973–1951=2333–2311=3251–3229=3491–3469=…, 24=1693–1669=2203–2179=4201–4177=4547–4523=,…, 26=2503–2477=3163–3137=3669–3643=5557–5531=…, 28=2999–2971=3299–3271=5147–5119=10037–10009=…., 30=4327–4297=4861–4831=5381–5351=5779–5749=,…so on.

 

REFERENCES:

1.      Goldbach, C. (1742) Letter to Euler, 7 June.

2.      Jagjit Singh, Prime numbers and Goldbach Conjecture, Research J. Science and Tech. 5(1):-Jan.-Mar. 2013, 120-122.

3.      Renyi, A. (1962). On the representation of an even number as the sum of a single prime and a single almost-prime number. Amer. Math. Soc. Transl. 19(2): 299-321.

4.      Schnirelmann, L. (1930) On additive properties of numbers. Izv. Donskowo Politechn. Inst. (Nowotscherkask), 14(2-3):3-28.

5.      Vinogradov, I. M.(1937) The representation of an odd number as sum of three primes. (Russian) Dokl. Akad. Nouk SSSR, 16: 139-142.

 

 

 

Received on 16.01.2014    Accepted on 01.02.2014

© EnggResearch.net All Right Reserved

Int. J. Tech. 4(1): Jan.-June. 2014; Page 67-70