**Prime Numbers and Goldbach
Conjecture**

**Dr Jagjit Singh**

Department of
Mathematics, Govt. College Bilaspur (H.P.)-174001,
India

*Corresponding Author: **jagjitsinghpatial@gmail.com**

**ABSTRACT:**

In this paper, two laws relating to prime numbers have
been given. An attempt has been made to prove the Goldbach’s
conjecture that every even natural number greater than or equal to 4 can be
expressed as a sum of two prime numbers. A result relating to counting of prime
and composite numbers within the range from 1 to n has also been presented
here.

**1. INTRODUCTION:**

There are
innumerable results and conjectures relating to the prime numbers 2, 3, 5, 7,
… …. A few conjectures concerning prime
numbers have been proved and many of them are still waiting for their proofs.
As much it is easy to conjecture about the prime numbers, so much is difficult
to give the proof of valid results of the prime numbers. One prominent
conjecture is of Goldbach [1]. Goldbach
conjectured that every even natural number greater than or equal to 4 can be
expressed as a sum of two prime numbers. The conjecture has been verified for
all even numbers less than 33×*n *from some point on is a sum of M or fewer primes:

*n*)

If this M
is known and it would be equal to 2 then this would prove Goldbach’s
conjecture. Yin Wen-Lin[6] proved that M≤18. Vinogradov[5] proved that for some point on every odd
number is the sum of three primes. Renyi[2] proved that there is a number M such that every
sufficiently large even number *n * can be expressed as a sum of a prime number
and another number which has no more than M prime factors.

**2.** **TWO LAWS OF PRIME NUMBERS**

**Theorem-1 (First law of prime numbers)**

If natural
numbers are divided into sets

If there
exists a prime number in set

Proof: let

**REFERENCES:**

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

2. 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.

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

4. Shen, Mok-Kong (1964) On checking the Goldbach conjecture. Nordisk Tidskr.
Informations –Behandling,
4: 243-245; MR 30, #3051.

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

6.
Yin, Wen-lin (1956) Note on the representation of large integers
as sums of primes. Bull. Acad. Polon. Sci. CI. III,
4: 793-795; MR 19, 16.

Received on 14.01.2013 Accepted on 09.02.2013

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*Research J. Science and Tech** 5(1): Jan.-Mar.2013** **page** 120-122*