Dr Jagjit Singh
Department of Mathematics, Govt. College Bilaspur (H.P.)-174001, India
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 . 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×
If this M is known and it would be equal to 2 then this would prove Goldbach’s conjecture. Yin Wen-Lin proved that M≤18. Vinogradov proved that for some point on every odd number is the sum of three primes. Renyi 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)
numbers are divided into sets
exists a prime number in set
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
©A&V Publications all right reserved