**Conjecture 61. THE
MEAN MAXIMUM GAP BETWEEN PRIMES**

Luis Rodríguez sent the following
conjecture:

p(n)= Prime

p(n+1)= Inmediate superior

Dp=p(n+1)-p(n)

Expected value of gap near p(n)=[Log(Pn)-LogLog(Pn)]^2

Z=Log p(n)

**CONJECTURE:**

Lim Sup{Dp/(Z - Log Z)^2}=1 when n --> Infinte

Luis sent the following Table on
support of his conjecture:

p(n) |
Gap |
Conjecture |
Chi^2 |
Relation |

1327 |
34 |
27.2 |
1.69 |
1.25 |

31397 |
72 |
64.3 |
0.92 |
1.1198 |

370261 |
112 |
105.5 |
0.4 |
1.0616 |

4652353 |
154 |
159.3 |
0.18 |
0.9667 |

20831323 |
210 |
196.8 |
0.89 |
1.067 |

47326693 |
220 |
219 |
0 |
1.0045 |

436273009 |
282 |
285.7 |
0.05 |
0.987 |

4302407359 |
354 |
364 |
0.27 |
0.9725 |

10726904659 |
382 |
398.2 |
0.66 |
0.9593 |

25056082087 |
456 |
431.3 |
1.41 |
1.0573 |

304599508537 |
514 |
536.7 |
0.96 |
0.9577 |

1408695493609 |
588 |
607.2 |
0.61 |
0.9684 |

2614941710599 |
652 |
637 |
0.35 |
1.0235 |

19581334192423 |
766 |
739 |
0.99 |
1.0365 |

42842283925351 |
778 |
780.8 |
0 |
0.9964 |

90874329411493 |
804 |
822 |
0.02 |
0.9781 |

218209405436543 |
906 |
871.4 |
1.37 |
1.0397 |

2764496039544377 |
990 |
1023 |
1.07 |
0.9677 |

7123663452897877 |
1044 |
1082.7 |
1.38 |
0.9643 |

19182559946240569 |
1120 |
1147.1 |
0.64 |
0.9763 |

1693182318746371 |
1132 |
992.8 |
19.52 |
To discard |

80873624627234849 |
1220 |
1244 |
0.46 |
0.9807 |

418032645936712127 |
1370 |
1359.5 |
0.08 |
1.0077 |

804212830686677669 |
1442 |
1407 |
0.87 |
1.0249 |

Chi^2 = 15.2. For 23 degrees of
liberty that means 90% of level of significance.

QUESTION: Can you design a formula more adjusted than this?

____

Ref. 'The New Book of Prime N. Records' - PAULO RIBENBOIM - Springer
1996

'First ocurrence of prime gaps' - NICELY & NYMAN (2005) from:
http://www.trnicely.net/gaps/

Contributions came from J K Andersen &
Jan van Delden

***

J. K. Andersen wrote:

The formulation in conjecture 61

Lim Sup{Dp/(Z - Log Z)^2}=1 when n --> Infinte

is equivalent to Cramér's conjecture (conjecture 7)

Lim Sup{Dp/Z^2}=1 when n --> Infinite

They are equivalent because:

{Dp/Z^2} / {Dp/(Z - Log Z)^2} = (Z - Log Z)^2 / Z^2

= 1 - (2*Log Z)/Z + (Log Z)^2/Z^2 --> 1 when Z --> infinite

The last column in Rodríguez' table only stays so close to 1 because

he writes "To discard" instead of the value 1.1402 for the remarkable

gap of 1132 at 1693182318746371.

That data point is included in the table for Cramér's Conjecture at

http://wvwright.net/

In the definition of Lim Sup, any data point is eventually "discarded".

***

Jan wrote:

The way the problem is posed is not
fully specified. I do understand he wants to state a conjecture
regarding the maximal gap between primes, in contrary to for instance
Cramérs conjecture regarding the average primegap. For instance I
presume only those p(n) are allowed for which the corresponding gap
d(p(n)) is increasing.

Secondly I'm not sure whether Luis Rodríguez wants to use the lim sup. I
get the impression he wants to state something about the fact that a
maximum gap is used and hence uses the lim sup. Where, considering his
table, an 'ordinary' limit would be more approriate. Which would perhaps
be more in line with the name given to the conjecture.

Lastly his Chi^2 calculation/interpretation is not very appropriate.
First it can only be applied if the listed (increasing) gaps can be
considered independent. If so, the value 15.2 with 23 degrees of
freedom, states that the probability that such a Chi-square value is
found (or larger) given the values of the maximal gaps in the table is
about 90%. Hence the deviation between the found values and the expected
values can be considered small (enough), given the large probability of
discovering such a deviation. [Another approach could be to construct a
confidence interval around 15.2 and check whether the value 0 (no
deviation) lies inside this interval].

***

Luis replied to Jan:

My conjecture differ from original
Cramer's only in the divisor: (ln p- ln ln p)^2 instead (ln p)^2.

It has been observed that the difference between Cramer and the real
maximum gap augment with p .

My curve gives little differences negatives and positives.

The use of Chi^2 is based in the supposition that the true law of max.
diff. is (ln p - ln ln p)^2,

then the differences between the gaps and the formula are experimental
errors.

The aim in searching a formula was the necessity of a mean for
predicting the gap near a large number or

to predict the number in whose neighbourhood a large gap can occur .

Example. The gap 2000 is expected to occur in the neighbourhood of 1.4 x
10^21.

***

Jan Feliksiak wrote on April 1,2021:

***