Mathematical Computation June 2013, Volume 2, Issue 2, PP.32-35
A Conjecture about Prime Maximal Gaps Wenlong Du School of Information Science and Engineering, Southeast University, Nanjing 210096, PR. China Email: duwenlong_25@126.com
Abstract This paper, in which the Cramér conjecture has been studied and the value of the Prime Maximal Gap and (logN)2 has been compared, presents a new conjecture about Prime Maximal Gaps. It has been confirmed that the value of the new conjecture is very close to that of the Prime Maximal Gaps. Keywords: Conjecture; Prime Number; Prime Maximal Gaps
1 INTRODUCTION As it is well known that prime number is 2,3,5 , thus all these prime number are denoted by p1 , p2 , , pn . The prime maximal gap max ( pn 1 pn ) means the maximum value of pn1 N
( p2 p1 , p3 p2 , , pn1 pn ) . The prime maximal gap max ( pn 1 pn ) , one of the most important prime pn1 N
properties, is the research topic of many scientists. The prime maximal gaps [2] are discovered when N is less than
4 1018 . In 1937, Cramér gave a conjecture [1] about the prime maximal that limsup( pn1 pn ) log pn 1 which 2
n
is still an unproven conjecture.
2 THE NEW CONJECTURE When n is finite, we compare the size of max pn 1 pn and log N . 2
pn1 N
TABLE 1 THE CRAMÉR CONJECTURE
Serial number
Natural number
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 132
2 3 7 23 89 113 523 887 1129 1327 9551 15683 19609 31397 155921 360653 370261 492113 1349533 1357201
Actual value
Theoretical value
Ratio
1 2 4 6 8 14 18 20 22 34 36 44 52 72 86 96 112 114 118
—— —— 4 10 20 22 39 46 49 52 84 93 98 107 143 164 164 172 199 199
—— —— 1.00 1.67 2.50 1.57 2.17 2.30 2.23 1.53 2.33 2.11 1.88 1.49 1.66 1.71 1.46 1.51 1.69 1.51
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2log log N log N —— —— 0.65 1.19 1.68 1.05 1.27 1.30 1.25 0.83 1.13 1.00 0.87 0.67 0.69 0.68 0.58 0.59 0.63 0.57