Cramérs Conjecture. Roupam Ghosh B.Sc.(Computer Science), 2007 University of Pune Pune, India.

Transcription

1 Cramérs Cojecture arxiv: v1 [math.gm] 3 Sep 010 Roupam Ghosh B.Sc.Computer Sciece, 007 Uiversity of Pue Pue, Idia roupam.ghosh@gmail.com September 7, 010 Abstract I this paper we prove the cojecture o prime gaps by Cramér. 1 Backgroud How thoroughly it is igraied i mathematical sciece that every real advace goes had i had with the ivetio of sharper tools ad simpler methods which, at the same time, assist i uderstadig earlier theories ad i castig aside some more complicated developmets. David Hilbert Harald Cramér gave some of the most ifluetial isights ito prime differeces. Cramér showed that assumig the truth of Riema Hypothesis, oe has +1 = O. He subsequetly cojectured i 1937 that +1 = O, ad more specifically that lim sup+1 / = 1 which is a much tighter boud, tha that implied by the Riema Hypothesis. Cramér s approach was based o statistical ad probabilistic grouds. These ew isights, brought i a set of ew problems. It became clear, that it was importat 1

2 to uderstad the extreme cases, alog with the average which had bee explored through the prime umber theorem. I 1931, Westzythius proved lim sup+1 / = This expressio +1 / is ow kow as the merit of a prime gap. As of August 009 the largest kow merit is So we have a log way to go before this merit reaches ifiity. This also idicates the possibility, that extreme cases i the study of prime gaps, are relatively rare. Recetly, i 005, aother spectacular breakthrough was made, whe D. A. Goldsto, J. Pitz ad C. Y. Yıldırım together showed, +1 / = 0 So, a few substatial breakthroughs were made i the theory of prime differeces, but still majority of the area remais uexplored. Most cases of exploratios i this uchartered territory was made usig heuristic ad probabilistic argumets, as was doe by Cramér himself. Cramér s Cojecture Theorem Cramér, I the big-oh otatio we have, d = O where, is the th prime ad d = +1 Proof. We ca see easily, that, p ie., logx d 1 Li+1 Li d where, Lix is the logarithmic itegral fuctio.

3 From which we get, p logx = Li+1 Lip p logx d p logx But we have, Hece, we get p p logx logp d p +1 logx +1 d lim sup lim sup logp Now, we have > p. Also, as, lim p = 1. Moreover, we kow from the prime umber theorem, lim +1 / = 1. Hece, we have Which gives, exp From which we get, exp ie., exp logp p = 1 +1 exp log sup exp log p +1 p logp sup exp +1 From the prime umber theorem we kow that lim / = 1, p logp exp1 sup exp

4 Takig logarithm we get, Hece, we get 1 sup d +1 lim sup lim sup Applyig Bertrad s postulate we get, Which gives us our required result, i.e., logp d lim sup log 1 d = O 6 logp Coclusio Mathematicias have tried i vai to this day to discover some order i the sequece of prime umbers, ad we have reaso to believe that it is a mystery ito which the huma mid will ever peetrate. Leohard Euler Every research paper is icomplete, i the sese that, there s a lot more that is left to be writte. I hope the reader has uderstood what I wated to express, regardig the usefuless of the tools preseted i this paper. 4 Ackowledgemets As always, my thaks goes to my mom, dad, ad sister for makig my life woderful, ad also, to my fried Craig Feistei. Refereces [1] Sodow, Joatha ad Weisstei, Eric W. Bertrad s Postulate. From MathWorld A Wolfram Web Resource. 4

5 [] Ribeboim, P. The New Book of Prime Number Records, 3rd Editio, Spriger, p. 53, 1996 [3] Cramér, H. O the Order of Magitude of the Differece betwee Cosecutive Prime Numbers. Acta Arith., 3-46, [4] Ribeboim, P. The New Book of Prime Number Records, 3rd Editio, Spriger, p. 5, 1996 [5] Graville, A. Harald Cramér ad the Distributio of Prime Numbers. Scad. Act. J. 1, 1-8,