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× . It has been found that every even number greater than 6 and less than 33×  is the sum of two distinct primes (Shen [4]).  Schnirelmann [3] proved that there is a number M such that every number n from some point on is a sum of M or fewer primes:

(For sufficient large 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 , ,… according to their order and some monotonically increasing sequence  of natural numbers so that first   natural numbers belong to    natural numbers belong to ,   natural numbers belong to  and proceeding in the same way,   natural numbers belong to  , therefore number of natural numbers in the preceding  set  is  , number of natural numbers in the succeeding set

If there exists a prime number in set  then there exists at least one prime number in the set .

Proof: let  contains a prime number. Numbers of natural numbers in is  which has been chosen according to the respective terms of the monotonically increasing sequence  with all its terms are natural numbers and  contains natural numbers according to the sequence . Suppose does not contain any prime number, because  is an arbitrary natural number and can be increased indefinitely and it can tend to infinity such that contains a prime number and does not contain a prime number, this implies that there are only finite number of prime number among the natural numbers but this contradicts the fact that prime numbers are infinite.  Therefore,  must contain a prime number. Also when  tends to infinity then  and become indefinitely large therefore if  having a prime number then  must have at least one prime number.

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. InformationsBehandling, 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