Primes pairs between ( (n/2) - ( log(n) )^2 ) and ( (n/2) + ( log(n) )^2 ) adding to even n

Primes pairs between

((n / 2)  −  ( log n) 2 )

and

((n / 2) + ( log n) 2 )

adding to even

n

n	Prime less than n / 2 are listed in decreasing order, primes from n / 2 are listed in increasing order. Distinct primes in pairs adding to n are surrounded by ■ (black squares); prime equal to n / 2 surrounded by □ (white squares).
10 1 (4 primes) (1.5 pairs) Y	2 ■3■ □5□ ■7■
10 2 (11 primes) (3 pairs) Y	■47■ 43 ■41■ 37 31 ■29■ ■53■ ■59■ 61 67 ■71■
10 3 (14 primes) (2 pairs) Y	499 ■491■ 487 ■479■ 467 463 461 457 503 ■509■ ■521■ 523 541 547
10 4 (23 primes) (1 pair) Y	4999 4993 4987 4973 4969 4967 4957 4951 4943 4937 4933 4931 ■4919■ 5003 5009 5011 5021 5023 5039 5051 5059 5077 ■5081■
10 5 (29 primes) (2 pairs) Y	49999 49993 49991 49957 49943 49939 49937 49927 49921 49919 49891 ■49877■ ■49871■ 50021 50023 50033 50047 50051 50053 50069 50077 50087 50093 50101 50111 50119 ■50123■ ■50129■ 50131
10 6 (27 primes) (1 pair) Y	499979 499973 499969 499957 ■499943■ 499927 499903 499897 499883 499879 499853 499819 500009 500029 500041 ■500057■ 500069 500083 500107 500111 500113 500119 500153 500167 500173 500177 500179
10 9 (41 primes) (2 pairs) Y	499999993 ■499999931■ 499999909 499999897 499999873 499999853 499999847 499999831 499999751 499999723 499999697 499999693 ■499999613■ 500000003 500000009 500000041 500000057 ■500000069■ 500000071 500000077 500000089 500000093 500000099 500000101 500000117 500000183 500000201 500000227 500000231 500000233 500000261 500000273 500000299 500000317 500000321 500000323 500000353 500000359 500000377 ■500000387■ 500000393
10 12 (52 primes) (2 pairs) Y	499999999979 499999999943 499999999901 499999999897 499999999847 499999999819 499999999799 ■499999999769■ 499999999739 499999999699 499999999661 499999999643 499999999571 499999999559 499999999511 499999999507 499999999501 499999999487 499999999451 499999999427 499999999403 499999999391 499999999357 ■499999999277■ 500000000023 500000000033 500000000089 500000000131 500000000147 500000000173 500000000191 500000000209 ■500000000231■ 500000000243 500000000263 500000000273 500000000333 500000000387 500000000413 500000000471 500000000509 500000000537 500000000551 500000000609 500000000611 500000000623 500000000627 500000000651 500000000677 ■500000000723■ 500000000737 500000000761
1477665855854 (2⋅738832927927) (48 primes) (1.5 pair) Y	738832927913 738832927909 738832927883 738832927879 738832927873 738832927867 738832927829 738832927799 738832927753 738832927711 738832927691 738832927681 738832927673 738832927663 738832927631 738832927607 738832927597 738832927579 738832927559 738832927543 738832927523 738832927499 738832927477 ■738832927387■ 738832927373 738832927367 738832927363 738832927331 738832927277 738832927271 738832927267 738832927253 738832927247 738832927201 738832927193 738832927187 738832927169 738832927153 □738832927927□<!-- gap of 540 (see A053303) from 738832927927 (see A053302) to 738832928467 --> ■738832928467■ 738832928473 738832928507 738832928569 738832928593 738832928621 738832928623 738832928629 738832928683

Cramér’s conjecture

Consider Cramér’s conjecture (by the Swedish mathematician Harald Cramér in 1936, yet unproved)

n → ∞lim supn → ∞  pn  +1 − pn (log pn ) 2 = 1,

where

pn

is the

n

-th prime.
In the [rare] cases where a primefree interval of width near

( log pn ) 2

falls somewhere in the middle of

• ((n / 2)  −  ( log n) 2 )

  to

  ((n / 2) + ( log n) 2 )

there will be a low probability of having prime pairs adding to even

n

.
In the [much rarer] cases where a primefree interval of width near

( log pn ) 2

almost coincides with either

• ((n / 2)  −  ( log n) 2 )

  to

  n / 2

  , or

• n / 2

  to

  ((n / 2) + ( log n) 2 )

  ,

there will be a near zero probability of having prime pairs adding to even

n

.
It thus appears that, asymptotically, it will happen infinitely often that there will be no pair of distinct primes within

((n / 2)  −  ( log n) 2 )

and

((n / 2) + ( log n) 2 )

adding to even

n

.

Question:

• Is the asymptotic density of cases where no pair of distinct primes within

  ((n / 2)  −  ( log n) 2 )

  and

  ((n / 2) + ( log n) 2 )

  adding to even

  n

  equal to 0?

Sequences

A053303 Length of maximal prime gap

pk  +1  −  pk

with starting prime

pk < 10 n

.

{4, 8, 20, 36, 72, 114, 154, 220, 282, 354, 464, 540, 674, 804, 906, 1132, ...}

A053302 Largest

n

-digit prime at the start of a record in the RECORDS transform of the prime gaps.

{7, 89, 887, 9551, 31397, 492113, 4652353, 47326693, 436273009, 4302407359, 42652618343, 738832927927, 7177162611713, 90874329411493, 218209405436543, 1693182318746371, ...}

External links

• Weisstein, Eric W., Cramér Conjecture, from MathWorld—A Wolfram Web Resource.
• Weisstein, Eric W., Cramér–Granville Conjecture, from MathWorld—A Wolfram Web Resource.
• Weisstein, Eric W., Shanks’ Conjecture (and Wolf  ’s conjecture), from MathWorld—A Wolfram Web Resource.
• Ahmad Sabihi, On the Firoozbakht’s conjecture, 2016. arXiv:1603.08917
• Luan Alberto Ferreira, Hugo Luiz Mariano, Prime gaps and the Firoozbakht Conjecture, 2018.