A conjecture about prime maximal gaps

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 pn1  N

( p2  p1 , p3  p2 , , pn1  pn ) . The prime maximal gap max ( pn 1  pn ) , one of the most important prime pn1  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( pn1  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

pn1  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

TABLE 1 THE CRAMÉR CONJECTURE (CONTINUE)

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

2010733 4652353 17051707 20831323 47326693 122164747 189695659 191912783 387096133 436273009 1294268491 1453168141 2300942549 3842610773 4302407359 10726904659 20678048297 22367084959 25056082087 42652618343 127976334671 182226896239 241160624143 297501075799 303371455241 304599508537 416608695821 461690510011 614487453523 738832927927 1346294310749 1408695493609 1968188556461 2614941710599 7177162611713 13829048559701 19581334192423 42842283925351 90874329411493 171231342420521 218209405436543 1189459969825483 1686994940955803 1693182318746371 43841547845541059 55350776431903243 80873624627234849 203986478517455989 218034721194214273 305405826521087869 352521223451364323 401429925999153707 418032645936712127 804212830686677669 1425172824437699411 1. Actual value:

max  pn 1  pn 

pn1  N

When N is large, it is found that max  pn 1  pn  pn1  N

148 154 180 210 220 222 234 248 250 282 288 292 320 336 354 382 384 394 456 464 468 474 486 490 500 514 516 532 534 540 582 588 602 652 674 716 766 778 804 806 906 916 924 1132 1184 1198 1220 1224 1248 1272 1328 1356 1370 1442 1476

 log N 

2

211 236 277 284 312 347 363 364 391 396 440 445 465 487 492 533 564 568 573 599 654 672 687 698 699 699 716 721 737 747 780 783 801 818 876 916 937 985 1033 1074 1090 1205 1229 1230 1468 1486 1516 1589 1594 1621 1632 1643 1646 1700 1747

2. Theoretical value:

1.43 1.53 1.54 1.35 1.42 1.56 1.55 1.47 1.56 1.40 1.53 1.52 1.45 1.45 1.39 1.40 1.47 1.44 1.26 1.29 1.40 1.42 1.41 1.42 1.40 1.36 1.39 1.36 1.38 1.38 1.34 1.33 1.33 1.25 1.30 1.28 1.22 1.27 1.28 1.33 1.20 1.32 1.33 1.09 1.24 1.24 1.24 1.30 1.28 1.27 1.23 1.21 1.20 1.18 1.18

 log N 

2

3. Ratio:

0.52 0.54 0.52 0.45 0.46 0.49 0.48 0.45 0.47 0.42 0.44 0.44 0.41 0.41 0.41 0.38 0.39 0.38 0.33 0.34 0.35 0.36 0.35 0.35 0.35 0.34 0.34 0.33 0.34 0.33 0.32 0.32 0.31 0.29 0.30 0.29 0.27 0.28 0.28 0.28 0.25 0.27 0.27 0.22 0.24 0.24 0.23 0.24 0.24 0.23 0.23 0.22 0.22 0.21 0.21

 log N 

2

max  pn 1  pn 

pn1  N

 2log log N log N  1 . So the conjecture in this paper is

max  pn1  pn   log N (log N  2log log N )  2  N  7  .

pn1  N

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TABLE 2 THE CRAMÉR CONJECTURE AND THE CONJECTURE IN THIS PAPER

Serial number

Natural number

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

2 3 7 23 89 113 523 887 1129 1327 9551 15683 19609 31397 155921 360653 370261 492113 1349533 1357201 2010733 4652353 17051707 20831323 47326693 122164747 189695659 191912783 387096133 436273009 1294268491 1453168141 2300942549 3842610773 4302407359 10726904659 20678048297 22367084959 25056082087 42652618343 127976334671 182226896239 241160624143 297501075799 303371455241 304599508537 416608695821 461690510011 614487453523 738832927927 1346294310749 1408695493609 1968188556461 2614941710599 7177162611713 13829048559701 19581334192423 42842283925351

Actual value 1 2 4 6 8 14 18 20 22 34 36 44 52 72 86 96 112 114 118 132 148 154 180 210 220 222 234 248 250 282 288 292 320 336 354 382 384 394 456 464 468 474 486 490 500 514 516 532 534 540 582 588 602 652 674 716 766 778

Theoretical value1 —— —— 3 5 9 10 18 22 24 25 45 51 54 61 86 100 101 106 127 127 135 154 186 191 213 240 253 253 275 279 314 318 334 352 357 390 416 419 423 445 490 505 518 527 528 528 542 547 560 568 596 598 614 628 678 711 729 771

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Ratio1 —— —— 0.75 0.83 1.13 0.71 1.00 1.10 1.09 0.74 1.25 1.16 1.04 0.85 1.00 1.04 0.90 0.93 1.08 0.96 0.91 1.00 1.03 0.91 0.97 1.08 1.08 1.02 1.10 0.99 1.09 1.09 1.04 1.05 1.01 1.02 1.08 1.06 0.93 0.96 1.05 1.07 1.07 1.08 1.06 1.03 1.05 1.03 1.05 1.05 1.02 1.02 1.02 0.96 1.01 0.99 0.95 0.99

Theoretical value2 —— —— 4 10 20 22 39 46 49 52 84 93 98 107 143 164 164 172 199 199 211 236 277 284 312 347 363 364 391 396 440 445 465 487 492 533 564 568 573 599 654 672 687 698 699 699 716 721 737 747 780 783 801 818 876 916 937 985

Ratio2 —— —— 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 1.43 1.53 1.54 1.35 1.42 1.56 1.55 1.47 1.56 1.40 1.53 1.52 1.45 1.45 1.39 1.40 1.47 1.44 1.26 1.29 1.40 1.42 1.41 1.42 1.40 1.36 1.39 1.36 1.38 1.38 1.34 1.33 1.33 1.25 1.30 1.28 1.22 1.27

TABLE 2 THE CRAMÉR CONJECTURE AND THE CONJECTURE IN THIS PAPER (CONTINUE)

59

90874329411493

804

812

1.01

1033

1.28

60

171231342420521

806

847

1.05

1074

1.33

61

218209405436543

906

861

0.95

1090

1.20

62 63 64

1189459969825483 1686994940955803 1693182318746371

916 924 1132

961 982 982

1.05 1.06 0.87

1205 1229 1230

1.32 1.33 1.09

65

43841547845541059

1184

1191

1.01

1468

1.24

66

55350776431903243

1198

1207

1.01

1486

1.24

67

80873624627234849

1220

1233

1.01

1516

1.24

68 69 70

203986478517455989 218034721194214273 305405826521087869

1224 1248 1272

1297 1301 1325

1.06 1.04 1.04

1589 1594 1621

1.30 1.28 1.27

71

352521223451364323

1328

1336

1.01

1632

1.23

72

401429925999153707

1356

1345

0.99

1643

1.21

73

418032645936712127

1370

1348

0.98

1646

1.20

74

804212830686677669

1442

1395

0.97

1700

1.18

75

1425172824437699411

1476

1437

0.97

1747

1.18

1. Actual value:

max  pn 1  pn 

pn1  N

3. Ratio 1:

2. Theoretical value 1:

log N (log N  2log log N )  2

log N (log N  2log log N )  2

4.Theoretical value 2:

 log N 

2

5. Ratio 2:

 log N 

2

max  pn1  pn 

pn1  N

max  pn 1  pn 

pn1  N

The conjecture in this paper gives an approximate value of the prime maximal gap. Which is close to the actual value, meaning a lot for finding the max( pn  pn 1 ) when N is larger than 4  1018 . pn  N

REFERENCES [1]

Cramér H. “On the order of magnitude of the difference between consecutive prime numbers.” Acta Arithmetica, 2(1936): 23–46

[2]

Thomas R N. “First occurrence prime gaps” (2013). URL: http://www.trnicely. net/gaps/gapsist.html

AUTHORS Du Wenlong was born in Hebei, China, in 1983. He received the B.S. degree in Physical Electronics from the North University of China in 2010. He is a Ph.D of Southeast University in China from 2012 to now. His major is integrated circuit design.

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