**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 2^{n} 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 2^{n} 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 couldnt 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, 2*n*≥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, 2*n*≥12 and .

**THEOREM-5:
**

Every
even number 2*n*≥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, 2*n*≥6 and . Let this one prime number is therefore or i.e. every even number 2*n*≥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.

83=5, 85=3; therefore number of primes obtained is 2 and number of composites is nil when prime numbers from 3 to 83 are subtracted from 8.

103=7, 105=5, 107=3; therefore number of primes obtained is 3 and number of composites is nil when prime numbers from 3 to 103 are subtracted from 10.

123=9, 125=7, 127=5; therefore number of primes obtained is 2 and number of composites is 1 when prime numbers from 3 to 123 are subtracted from 12.

143=11, 145=9, 147=7, 1411=3; therefore number of primes obtained is 3 and number of composites is 1 when prime numbers from 3 to 143 are subtracted from 14.

163=13, 165=11, 167=9, 1611=5, 1613=3; therefore number of primes obtained is 4 and number of composites is 1 when prime numbers from 3 to 163 are subtracted from 16.

183=15, 185=13, 187=11, 1811=7, 1813=5; therefore number of primes obtained is 4 and number of composites is 1 when prime numbers from 3 to 183 are subtracted from 18.

203=17, 205=15, 207=13, 2011=9, 2013=7, 2017=3; therefore number of primes obtained is 4 and number of composites is 2 when prime numbers from 3 to 203 are subtracted from 20.

223=19, 225=17, 227=15, 2211=11, 2213=9, 2217=5, 2219=3; therefore number of primes obtained is 5 and number of composites is 2 when prime numbers from 3 to 223 are subtracted from 22.

243=21, 245=19, 247=17, 2411=13, 2413=11, 2417=7, 2419=5; therefore number of primes obtained is 6 and number of composites is 1 when prime numbers from 3 to 243 are subtracted from 24.

263=23, 265=21, 267=19, 2611=15, 2613=13, 2617=9, 2619=7, 2623=3; therefore number of primes obtained is 5 and number of composites is 3 when prime numbers from 3 to 263 are subtracted from 26.

283=25, 285=23, 287=21, 2811=17, 2813=15, 2817=11, 2819=9, 2823=5; therefore number of primes obtained is 4 and number of composites is 4 when prime numbers from 3 to 283 are subtracted from 28.

303=27, 305=25, 307=23, 3011=19, 3013=17, 3017=13, 3019=11, 3023=7; therefore number of primes obtained is 6 and number of composites is 2 when prime numbers from 3 to 303 are subtracted from 30.

323=29, 325=27, 327=25, 3211=21, 3213=19, 3217=15, 3219=13, 3223=9, 3229=3; therefore number of primes obtained is 4 and number of composites is 5 when prime numbers from 3 to 323 are subtracted from 32.

343=31, 345=29, 347=27, 3411=23, 3413=21, 3417=17, 3419=15, 3423=11, 3429=5, 3431=3; therefore number of primes obtained is 7 and number of composites is 3 when prime numbers from 3 to 343 are subtracted from 34.

363=33, 365=31, 367=29, 3611=25, 3613=23, 3617=19, 3619=17, 3623=13, 3629=7, 3631=5; therefore number of primes obtained is 8 and number of composites is 2 when prime numbers from 3 to 363 are subtracted from 36.

383=35, 385=33, 387=31, 3811=27, 3813=25, 3817=21, 3819=19, 3823=15, 3829=9, 3831=7; therefore number of primes obtained is 3 and number of composites is 7 when prime numbers from 3 to 383 are subtracted from 38.

403=37, 405=35, 407=33, 4011=29, 4013=27, 4017=23, 4019=21, 4023=17, 4029=11, 4031=9, 4037=3; therefore number of primes obtained is 6 and number of composites is 5 when prime numbers from 3 to 403 are subtracted from 40.

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

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

**Table-1**

Even
number=2 |
Number of prime |
Number of composite |
Even
number=2 |
Number of prime |
Number of composite |
Even
number=2 |
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 53=2, 117=4, 29 23=6, 9789=8, 149 139=10, 211199=12, 127113=14,18471831=16, 541523=18, 907887=20, 19731951=22, 16931669=24, 25032477=26, 29992971=28, 43274297=30 56235591=32, 13611327=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=53=75=1311=1917= , 4=117=1713=2319=4137= , 6=2923=3731=5347=5953= , 8=9789=367359=387389=409401= , 10=149139=191181=251241, 293283= , 12=211199=223211=479467=521509= , 14=127113=307293=331317=787773= , 16=18471831=19491933=21292113=22372221= , 18=10871069=12771259=13991381=17771759= , 20=907887=16571637=31093089=34333413=57375717= ,22=11511129=19731951=23332311=32513229=34913469= , 24=16931669=22032179=42014177=45474523=, , 26=25032477=31633137=36693643=55575531= , 28=29992971=32993271=51475119=1003710009= ., 30=43274297=48614831=53815351=57795749=, 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) |